Theorem cbvmpo 7483

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 2891	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2891	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpo.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpo.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpo.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpo.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2730	. 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 7482	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnfc 2876   ∈ cmpo 7389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5170  df-oprab 7391  df-mpo 7392

This theorem is referenced by:  fvmpopr2d  7551  el2mpocsbcl  8064  fnmpoovd  8066  fmpoco  8074  mpocurryd  8248  fvmpocurryd  8250  xpf1o  9103  cnfcomlem  9652  fseqenlem1  9977  relexpsucnnr  14991  gsumdixp  20228  evlslem4  21983  madugsum  22530  cnmpt2t  23560  cnmptk2  23573  fmucnd  24179  fsum2cn  24762  aks6d1c7lem3  42170  fmpocos  42222  fmuldfeqlem1  45580  smflim  46775