Theorem cbvmpt2v 6111

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4259, 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 e. A , y e. B |-> C ) = ( z e. A , w e. 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 2540	. 2 |- F/_ z C
2		nfcv 2540	. 2 |- F/_ w C
3		nfcv 2540	. 2 |- F/_ x D
4		nfcv 2540	. 2 |- F/_ y D
5		cbvmpt2v.1	. . 3 |- ( x = z -> C = E )
6		cbvmpt2v.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2456	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpt2 6110	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1649   e. cmpt2 6042

This theorem is referenced by:  seqomlem0  6665  dffi3  7394  cantnfsuc  7581  fin23lem33  8181  om2uzrdg  11251  uzrdgsuci  11255  sadcp1  12922  smupp1  12947  imasvscafn  13717  sylow1  15192  sylow2b  15212  sylow3lem5  15220  sylow3  15222  efgmval  15299  efgtf  15309  txbas  17552  bcth  19235  opnmbl  19447  mbfimaopn  19501  mbfi1fseq  19566  opsqrlem3  23598  dya2iocival  24576  sxbrsigalem5  24591  sxbrsigalem6  24592  cvmliftlem15  24938  cvmlift2  24956  mblfinlem  26143  sdc  26338  tendoplcbv  31257  dvhvaddcbv  31572  dvhvscacbv  31581

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-oprab 6044  df-mpt2 6045