Theorem cbvmpt2 5603

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 2219	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2219	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpt2.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpt2.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpt2.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpt2.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2082	. 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 5602	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284  Ⅎwnfc 2206   ↦ cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  cbvmpt2v  5604  fmpt2co  5857