Theorem cbvmpt2v 5584

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3851, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpt2v.1	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
cbvmpt2v.2	⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)

Assertion

Ref	Expression
cbvmpt2v	⊢ (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 cbvmpt2v

Step	Hyp	Ref	Expression
1		nfcv 2178	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2178	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2178	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2178	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpt2v.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpt2v.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2092	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpt2 5583	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1243   ↦ cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  frec2uzrdg  9195  frecuzrdgsuc  9201  resqrexlemfp1  9607  resqrex  9624