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Theorem xpsfrn2 16542
 Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
Assertion
Ref Expression
xpsfrn2 ((𝐴𝑉𝐵𝑊) → ran 𝐹 = X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘))
Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝑘,𝑉   𝐵,𝑘,𝑥,𝑦   𝑘,𝑊
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑘)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem xpsfrn2
StepHypRef Expression
1 xpscfv 16534 . . . 4 ((𝐴𝑉𝐵𝑊𝑘 ∈ 2𝑜) → (({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
213expa 1148 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝑘 ∈ 2𝑜) → (({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
32ixpeq2dva 8161 . 2 ((𝐴𝑉𝐵𝑊) → X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘) = X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
4 xpsff1o.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
54xpsfrn 16541 . 2 ran 𝐹 = X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
63, 5syl6reqr 2850 1 ((𝐴𝑉𝐵𝑊) → ran 𝐹 = X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∅c0 4113  ifcif 4275  {csn 4366  ◡ccnv 5309  ran crn 5311  ‘cfv 6099  (class class class)co 6876   ↦ cmpt2 6878  2𝑜c2o 7791  Xcixp 8146   +𝑐 ccda 9275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-er 7980  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-cda 9276 This theorem is referenced by: (None)
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Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsfrn2 Structured version   Visualization version   GIF version

Theorem xpsfrn2 16542
 Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
Assertion
Ref Expression
xpsfrn2 ((𝐴𝑉𝐵𝑊) → ran 𝐹 = X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘))
Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝑘,𝑉   𝐵,𝑘,𝑥,𝑦   𝑘,𝑊
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑘)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem xpsfrn2
StepHypRef Expression
1 xpscfv 16534 . . . 4 ((𝐴𝑉𝐵𝑊𝑘 ∈ 2𝑜) → (({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
213expa 1148 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝑘 ∈ 2𝑜) → (({𝐴} +𝑐 {𝐵})‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
32ixpeq2dva 8161 . 2 ((𝐴𝑉𝐵𝑊) → X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘) = X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
4 xpsff1o.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
54xpsfrn 16541 . 2 ran 𝐹 = X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
63, 5syl6reqr 2850 1 ((𝐴𝑉𝐵𝑊) → ran 𝐹 = X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∅c0 4113  ifcif 4275  {csn 4366  ◡ccnv 5309  ran crn 5311  ‘cfv 6099  (class class class)co 6876   ↦ cmpt2 6878  2𝑜c2o 7791  Xcixp 8146   +𝑐 ccda 9275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-er 7980  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-cda 9276 This theorem is referenced by: (None)
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The text passage is a formal mathematical proof from the Metamath Proof Explorer, which is a project that aims to develop a complete, self-contained, and rigorously formalized proof of the main theorems of mathematics using a computer program. The passage contains a theorem, its description, hypotheses, and the proof steps.

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