Theorem cbvmpt2 6632

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

Hypotheses

Ref	Expression
cbvmpt2.1	⊢ Ⅎ𝑧𝐶
cbvmpt2.2	⊢ Ⅎ𝑤𝐶
cbvmpt2.3	⊢ Ⅎ𝑥𝐷
cbvmpt2.4	⊢ Ⅎ𝑦𝐷
cbvmpt2.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvmpt2	⊢ (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 cbvmpt2

Step	Hyp	Ref	Expression
1		nfcv 2751	. 2 ⊢ Ⅎ𝑧𝐵
2		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐵
3		cbvmpt2.1	. 2 ⊢ Ⅎ𝑧𝐶
4		cbvmpt2.2	. 2 ⊢ Ⅎ𝑤𝐶
5		cbvmpt2.3	. 2 ⊢ Ⅎ𝑥𝐷
6		cbvmpt2.4	. 2 ⊢ Ⅎ𝑦𝐷
7		eqidd 2611	. 2 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐵)
8		cbvmpt2.5	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpt2x 6631	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnfc 2738   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  cbvmpt2v  6633  el2mpt2csbcl  7137  fnmpt2ovd  7139  fmpt2co  7147  mpt2curryd  7282  fvmpt2curryd  7284  xpf1o  8007  cnfcomlem  8479  fseqenlem1  8730  relexpsucnnr  13613  gsumdixp  18432  evlslem4  19329  madugsum  20268  cnmpt2t  21286  cnmptk2  21299  fmucnd  21906  fsum2cn  22482  fmuldfeqlem1  38649  smflim  39663