Theorem cbvmpt2 5743

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)

Hypotheses

Ref	Expression
cbvmpt2.1	⊢ Ⅎ𝑧𝐶
cbvmpt2.2	⊢ Ⅎ𝑤𝐶
cbvmpt2.3	⊢ Ⅎ𝑥𝐷
cbvmpt2.4	⊢ Ⅎ𝑦𝐷
cbvmpt2.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvmpt2	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

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

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

Proof of Theorem cbvmpt2

Step	Hyp	Ref	Expression
1		nfcv 2229	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2229	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpt2.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpt2.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpt2.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpt2.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2090	. 2 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐵)
8		cbvmpt2.5	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpt2x 5742	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290  Ⅎwnfc 2216   ↦ cmpt2 5670

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-opab 3908  df-oprab 5672  df-mpt2 5673

This theorem is referenced by:  cbvmpt2v  5744  fmpt2co  5997  xpf1o  6616