Theorem cbvmpo 6110

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

Hypotheses

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

Assertion

Ref	Expression
cbvmpo	⊢ (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 cbvmpo

Step	Hyp	Ref	Expression
1		nfcv 2375	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2375	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpo.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpo.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpo.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpo.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2232	. 2 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐵)
8		cbvmpo.5	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpox 6109	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  Ⅎwnfc 2362   ∈ cmpo 6030

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-oprab 6032  df-mpo 6033

This theorem is referenced by:  cbvmpov  6111  fvmpopr2d  6168  fnmpoovd  6389  fmpoco  6390  xpf1o  7073  cnmpt2t  15087