Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconstmpt Structured version   Visualization version   GIF version

Theorem fconstmpt 5197
 Description: Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
Assertion
Ref Expression
fconstmpt (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem fconstmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 velsn 4226 . . . 4 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
21anbi2i 730 . . 3 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
32opabbii 4750 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-xp 5149 . 2 (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})}
5 df-mpt 4763 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
63, 4, 53eqtr4i 2683 1 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {csn 4210  {copab 4745   ↦ cmpt 4762   × cxp 5141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211  df-opab 4746  df-mpt 4763  df-xp 5149 This theorem is referenced by:  fconst  6129  fcoconst  6441  fmptsn  6474  fconstmpt2  6797  ofc12  6964  caofinvl  6966  xpexgALT  7203  cantnf  8628  cnfcom2lem  8636  repsconst  13565  harmonic  14635  geomulcvg  14651  vdwlem8  15739  ramcl  15780  pwsvscafval  16201  setcepi  16785  diag2  16932  pws0g  17373  0frgp  18238  pwsgsum  18424  lmhmvsca  19093  rrgsupp  19339  psrlinv  19445  psrlidm  19451  psrridm  19452  psrass23l  19456  psrass23  19458  mplcoe1  19513  mplcoe3  19514  mplcoe5  19516  mplmon2  19541  evlslem2  19560  evlslem1  19563  coe1z  19681  coe1mul2lem1  19685  coe1tm  19691  coe1sclmul  19700  coe1sclmul2  19702  evls1sca  19736  evl1sca  19746  uvcresum  20180  grpvrinv  20250  mdetunilem9  20474  pttoponconst  21448  cnmptc  21513  cnmptkc  21530  pt1hmeo  21657  tmdgsum2  21947  tsms0  21992  tgptsmscls  22000  resspwsds  22224  imasdsf1olem  22225  nmoeq0  22587  idnghm  22594  rrxcph  23226  ovolctb  23304  ovoliunnul  23321  vitalilem4  23425  vitalilem5  23426  ismbf  23442  mbfconst  23447  mbfss  23458  mbfmulc2re  23460  mbfneg  23462  mbfmulc2  23475  itg11  23503  itg2const  23552  itg2mulclem  23558  itg2mulc  23559  itg2monolem1  23562  itg0  23591  itgz  23592  itgvallem3  23597  iblposlem  23603  i1fibl  23619  itgitg1  23620  itgge0  23622  iblconst  23629  itgconst  23630  itgfsum  23638  iblmulc2  23642  itgmulc2lem1  23643  bddmulibl  23650  dvcmulf  23753  dvexp  23761  dvexp2  23762  dvmptid  23765  dvmptc  23766  dvef  23788  rolle  23798  dv11cn  23809  ftc1lem4  23847  ftc2  23852  tdeglem4  23865  ply1nzb  23927  plyconst  24007  plyeq0lem  24011  plypf1  24013  coeeulem  24025  plyco  24042  0dgr  24046  0dgrb  24047  dgrcolem2  24075  dgrco  24076  plyremlem  24104  elqaalem3  24121  iaa  24125  taylply2  24167  itgulm  24207  amgmlem  24761  lgam1  24835  ftalem7  24850  basellem8  24859  dchrfi  25025  dchrptlem3  25036  bra0  28937  padct  29625  xrge0mulc1cn  30115  esumnul  30238  esum0  30239  esumcvg  30276  ofcc  30296  mbfmcst  30449  sibf0  30524  0rrv  30641  ccatmulgnn0dir  30747  plymul02  30751  plymulx  30753  txsconnlem  31348  cvmliftphtlem  31425  faclim  31758  matunitlindflem1  33535  poimirlem30  33569  ovoliunnfl  33581  voliunnfl  33583  volsupnfl  33584  itg2addnclem  33591  iblmulc2nc  33605  itgmulc2nclem1  33606  itgmulc2nclem2  33607  itgmulc2nc  33608  itgabsnc  33609  bddiblnc  33610  ftc1cnnclem  33613  ftc1anclem3  33617  ftc1anclem5  33619  ftc1anclem7  33621  ftc1anclem8  33622  ftc2nc  33624  repwsmet  33763  rrnequiv  33764  mzpconstmpt  37620  mzpcompact2lem  37631  mendlmod  38080  mendassa  38081  expgrowthi  38849  expgrowth  38851  binomcxplemrat  38866  binomcxplemnotnn0  38872  rnmptc  39667  climconstmpt  40208  iblconstmpt  40489  iblempty  40499  itgiccshift  40514  itgperiod  40515  itgsbtaddcnst  40516  stoweidlem21  40556  lindsrng01  42582  aacllem  42875  amgmlemALT  42877
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Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconstmpt Structured version   Visualization version   GIF version

Theorem fconstmpt 5197
 Description: Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
Assertion
Ref Expression
fconstmpt (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem fconstmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 velsn 4226 . . . 4 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
21anbi2i 730 . . 3 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
32opabbii 4750 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-xp 5149 . 2 (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})}
5 df-mpt 4763 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
63, 4, 53eqtr4i 2683 1 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {csn 4210  {copab 4745   ↦ cmpt 4762   × cxp 5141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211  df-opab 4746  df-mpt 4763  df-xp 5149 This theorem is referenced by:  fconst  6129  fcoconst  6441  fmptsn  6474  fconstmpt2  6797  ofc12  6964  caofinvl  6966  xpexgALT  7203  cantnf  8628  cnfcom2lem  8636  repsconst  13565  harmonic  14635  geomulcvg  14651  vdwlem8  15739  ramcl  15780  pwsvscafval  16201  setcepi  16785  diag2  16932  pws0g  17373  0frgp  18238  pwsgsum  18424  lmhmvsca  19093  rrgsupp  19339  psrlinv  19445  psrlidm  19451  psrridm  19452  psrass23l  19456  psrass23  19458  mplcoe1  19513  mplcoe3  19514  mplcoe5  19516  mplmon2  19541  evlslem2  19560  evlslem1  19563  coe1z  19681  coe1mul2lem1  19685  coe1tm  19691  coe1sclmul  19700  coe1sclmul2  19702  evls1sca  19736  evl1sca  19746  uvcresum  20180  grpvrinv  20250  mdetunilem9  20474  pttoponconst  21448  cnmptc  21513  cnmptkc  21530  pt1hmeo  21657  tmdgsum2  21947  tsms0  21992  tgptsmscls  22000  resspwsds  22224  imasdsf1olem  22225  nmoeq0  22587  idnghm  22594  rrxcph  23226  ovolctb  23304  ovoliunnul  23321  vitalilem4  23425  vitalilem5  23426  ismbf  23442  mbfconst  23447  mbfss  23458  mbfmulc2re  23460  mbfneg  23462  mbfmulc2  23475  itg11  23503  itg2const  23552  itg2mulclem  23558  itg2mulc  23559  itg2monolem1  23562  itg0  23591  itgz  23592  itgvallem3  23597  iblposlem  23603  i1fibl  23619  itgitg1  23620  itgge0  23622  iblconst  23629  itgconst  23630  itgfsum  23638  iblmulc2  23642  itgmulc2lem1  23643  bddmulibl  23650  dvcmulf  23753  dvexp  23761  dvexp2  23762  dvmptid  23765  dvmptc  23766  dvef  23788  rolle  23798  dv11cn  23809  ftc1lem4  23847  ftc2  23852  tdeglem4  23865  ply1nzb  23927  plyconst  24007  plyeq0lem  24011  plypf1  24013  coeeulem  24025  plyco  24042  0dgr  24046  0dgrb  24047  dgrcolem2  24075  dgrco  24076  plyremlem  24104  elqaalem3  24121  iaa  24125  taylply2  24167  itgulm  24207  amgmlem  24761  lgam1  24835  ftalem7  24850  basellem8  24859  dchrfi  25025  dchrptlem3  25036  bra0  28937  padct  29625  xrge0mulc1cn  30115  esumnul  30238  esum0  30239  esumcvg  30276  ofcc  30296  mbfmcst  30449  sibf0  30524  0rrv  30641  ccatmulgnn0dir  30747  plymul02  30751  plymulx  30753  txsconnlem  31348  cvmliftphtlem  31425  faclim  31758  matunitlindflem1  33535  poimirlem30  33569  ovoliunnfl  33581  voliunnfl  33583  volsupnfl  33584  itg2addnclem  33591  iblmulc2nc  33605  itgmulc2nclem1  33606  itgmulc2nclem2  33607  itgmulc2nc  33608  itgabsnc  33609  bddiblnc  33610  ftc1cnnclem  33613  ftc1anclem3  33617  ftc1anclem5  33619  ftc1anclem7  33621  ftc1anclem8  33622  ftc2nc  33624  repwsmet  33763  rrnequiv  33764  mzpconstmpt  37620  mzpcompact2lem  37631  mendlmod  38080  mendassa  38081  expgrowthi  38849  expgrowth  38851  binomcxplemrat  38866  binomcxplemnotnn0  38872  rnmptc  39667  climconstmpt  40208  iblconstmpt  40489  iblempty  40499  itgiccshift  40514  itgperiod  40515  itgsbtaddcnst  40516  stoweidlem21  40556  lindsrng01  42582  aacllem  42875  amgmlemALT  42877
 Copyright terms: Public domain W3C validator
user
The provided text is a formal mathematical proof from the Metamath Proof Explorer, specifically for the theorem `fconstmpt`. It describes the representation of a constant function using the mapping operation. The proof is structured with steps, hypotheses, and references to other theorems and axioms.

TYPE: NO_QUESTION

The text does not contain any mathematical questions or answers to questions. It is a formal proof of a theorem.