Theorem cbvmpt2v 6611

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4671, 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 2750	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2750	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2750	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2750	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpt2v.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpt2v.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2663	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpt2 6610	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1474   ↦ cmpt2 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-oprab 6531  df-mpt2 6532

This theorem is referenced by:  seqomlem0  7409  dffi3  8198  cantnfsuc  8428  fin23lem33  9028  om2uzrdg  12575  uzrdgsuci  12579  sadcp1  14964  smupp1  14989  imasvscafn  15969  mgmnsgrpex  17190  sgrpnmndex  17191  sylow1  17790  sylow2b  17810  sylow3lem5  17818  sylow3  17820  efgmval  17897  efgtf  17907  frlmphl  19887  pmatcollpw3lem  20355  mp2pm2mplem3  20380  txbas  21128  bcth  22879  opnmbl  23121  mbfimaopn  23174  mbfi1fseq  23239  motplusg  25183  ttgval  25501  numclwwlk5  26433  opsqrlem3  28219  mdetpmtr12  29053  madjusmdetlem4  29058  fvproj  29061  dya2iocival  29496  sxbrsigalem5  29511  sxbrsigalem6  29512  eulerpart  29605  sseqp1  29618  cvmliftlem15  30368  cvmlift2  30386  opnmbllem0  32439  mblfinlem1  32440  mblfinlem2  32441  sdc  32534  tendoplcbv  34905  dvhvaddcbv  35220  dvhvscacbv  35229  fsovcnvlem  37151  ntrneibex  37215  ioorrnopn  39025  hoidmvle  39314  ovnhoi  39317  hoimbl  39345  smflimlem6  39486  av-numclwwlk5  41564  funcrngcsetc  41812  funcrngcsetcALT  41813  funcringcsetc  41849  lmod1zr  42098