Theorem cbvmpov 6024

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4138, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpov.1	|- ( x = z -> C = E )
cbvmpov.2	|- ( y = w -> E = D )

Assertion

Ref	Expression
cbvmpov	|- ( 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 cbvmpov

Step	Hyp	Ref	Expression
1		nfcv 2347	. 2 |- F/_ z C
2		nfcv 2347	. 2 |- F/_ w C
3		nfcv 2347	. 2 |- F/_ x D
4		nfcv 2347	. 2 |- F/_ y D
5		cbvmpov.1	. . 3 |- ( x = z -> C = E )
6		cbvmpov.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2257	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpo 6023	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1372   e. cmpo 5945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-oprab 5947  df-mpo 5948

This theorem is referenced by:  frec2uzrdg  10552  frecuzrdgsuc  10557  iseqvalcbv  10602  resqrexlemfp1  11291  resqrex  11308  sqne2sq  12470  ennnfonelemnn0  12764  nninfdc  12795  txbas  14701  xmetxp  14950  mpomulcn  15009