Theorem cbvmpo 7544

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 2908	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpo.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpo.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpo.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpo.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2741	. 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 7543	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  Ⅎwnfc 2893   ∈ cmpo 7450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-oprab 7452  df-mpo 7453

This theorem is referenced by:  fvmpopr2d  7612  el2mpocsbcl  8126  fnmpoovd  8128  fmpoco  8136  mpocurryd  8310  fvmpocurryd  8312  xpf1o  9205  cnfcomlem  9768  fseqenlem1  10093  relexpsucnnr  15074  gsumdixp  20342  evlslem4  22123  madugsum  22670  cnmpt2t  23702  cnmptk2  23715  fmucnd  24322  fsum2cn  24914  aks6d1c7lem3  42139  fmpocos  42229  fmuldfeqlem1  45503  smflim  46698