Theorem cbvmpov 5999

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4125, 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 2336	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2336	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2336	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2336	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpov.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpov.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2246	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpo 5998	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1364   ∈ cmpo 5921

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-oprab 5923  df-mpo 5924

This theorem is referenced by:  frec2uzrdg  10483  frecuzrdgsuc  10488  iseqvalcbv  10533  resqrexlemfp1  11156  resqrex  11173  sqne2sq  12318  ennnfonelemnn0  12582  nninfdc  12613  txbas  14437  xmetxp  14686  mpomulcn  14745