Theorem cbvmpo 7452

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 2898	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2898	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpo.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpo.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpo.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpo.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2737	. 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 7451	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  Ⅎwnfc 2883   ∈ cmpo 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-opab 5161  df-oprab 7362  df-mpo 7363

This theorem is referenced by:  fvmpopr2d  7520  el2mpocsbcl  8027  fnmpoovd  8029  fmpoco  8037  mpocurryd  8211  fvmpocurryd  8213  xpf1o  9067  cnfcomlem  9608  fseqenlem1  9934  relexpsucnnr  14948  gsumdixp  20254  evlslem4  22031  madugsum  22587  cnmpt2t  23617  cnmptk2  23630  fmucnd  24235  fsum2cn  24818  aks6d1c7lem3  42436  fmpocos  42490  fmuldfeqlem1  45828  smflim  47021