Theorem cbvmpt2v 6633

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4677, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpt2v.1	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
cbvmpt2v.2	⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)

Assertion

Ref	Expression
cbvmpt2v	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑧   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpt2v

Step	Hyp	Ref	Expression
1		nfcv 2751	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2751	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2751	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpt2v.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpt2v.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2664	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpt2 6632	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1475   ↦ 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:  seqomlem0  7431  dffi3  8220  cantnfsuc  8450  fin23lem33  9050  om2uzrdg  12617  uzrdgsuci  12621  sadcp1  15015  smupp1  15040  imasvscafn  16020  mgmnsgrpex  17241  sgrpnmndex  17242  sylow1  17841  sylow2b  17861  sylow3lem5  17869  sylow3  17871  efgmval  17948  efgtf  17958  frlmphl  19939  pmatcollpw3lem  20407  mp2pm2mplem3  20432  txbas  21180  bcth  22934  opnmbl  23176  mbfimaopn  23229  mbfi1fseq  23294  motplusg  25237  ttgval  25555  numclwwlk5  26639  opsqrlem3  28385  mdetpmtr12  29219  madjusmdetlem4  29224  fvproj  29227  dya2iocival  29662  sxbrsigalem5  29677  sxbrsigalem6  29678  eulerpart  29771  sseqp1  29784  cvmliftlem15  30534  cvmlift2  30552  opnmbllem0  32615  mblfinlem1  32616  mblfinlem2  32617  sdc  32710  tendoplcbv  35081  dvhvaddcbv  35396  dvhvscacbv  35405  fsovcnvlem  37327  ntrneibex  37391  ioorrnopn  39201  hoidmvle  39490  ovnhoi  39493  hoimbl  39521  smflimlem6  39662  av-numclwwlk5  41542  funcrngcsetc  41790  funcrngcsetcALT  41791  funcringcsetc  41827  lmod1zr  42076