Theorem cbvmpt2v 6735

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4749, 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 2764	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2764	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2764	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2764	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpt2v.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpt2v.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2676	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpt2 6734	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1483   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  seqomlem0  7544  dffi3  8337  cantnfsuc  8567  fin23lem33  9167  om2uzrdg  12755  uzrdgsuci  12759  sadcp1  15177  smupp1  15202  imasvscafn  16197  mgmnsgrpex  17418  sgrpnmndex  17419  sylow1  18018  sylow2b  18038  sylow3lem5  18046  sylow3  18048  efgmval  18125  efgtf  18135  frlmphl  20120  pmatcollpw3lem  20588  mp2pm2mplem3  20613  txbas  21370  bcth  23126  opnmbl  23370  mbfimaopn  23423  mbfi1fseq  23488  motplusg  25437  ttgval  25755  opsqrlem3  29001  mdetpmtr12  29891  madjusmdetlem4  29896  fvproj  29899  dya2iocival  30335  sxbrsigalem5  30350  sxbrsigalem6  30351  eulerpart  30444  sseqp1  30457  cvmliftlem15  31280  cvmlift2  31298  opnmbllem0  33445  mblfinlem1  33446  mblfinlem2  33447  sdc  33540  tendoplcbv  36063  dvhvaddcbv  36378  dvhvscacbv  36387  fsovcnvlem  38307  ntrneibex  38371  ioorrnopn  40525  hoidmvle  40814  ovnhoi  40817  hoimbl  40845  smflimlem6  40984  funcrngcsetc  41998  funcrngcsetcALT  41999  funcringcsetc  42035  lmod1zr  42282