Theorem cbvmpo 7504

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 2925	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2925	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpo.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpo.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpo.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpo.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2764	. 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 7503	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  Ⅎwnfc 2910   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  fvmpopr2d  7572  el2mpocsbcl  8076  fnmpoovd  8078  fmpoco  8086  mpocurryd  8261  fvmpocurryd  8263  xpf1o  9123  cnfcomlem  9664  fseqenlem1  10004  relexpsucnnr  15058  gsumdixp  20396  evlslem4  22227  madugsum  22800  cnmpt2t  23830  cnmptk2  23843  fmucnd  24448  fsum2cn  25030  aks6d1c7lem3  42969  fmpocos  43024  fmuldfeqlem1  46318  smflim  47511