Theorem cbvmpov 5945

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4093, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpov.1	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
cbvmpov.2	⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)

Assertion

Ref	Expression
cbvmpov	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑧   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpov

Step	Hyp	Ref	Expression
1		nfcv 2317	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2317	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2317	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2317	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpov.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpov.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2228	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpo 5944	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1353   ∈ cmpo 5867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-opab 4060  df-oprab 5869  df-mpo 5870

This theorem is referenced by:  frec2uzrdg  10379  frecuzrdgsuc  10384  iseqvalcbv  10427  resqrexlemfp1  10986  resqrex  11003  sqne2sq  12144  ennnfonelemnn0  12390  nninfdc  12421  txbas  13329  xmetxp  13578