Theorem cbvmpov 5951

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4097, 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 2319	. 2 |- F/_ z C
2		nfcv 2319	. 2 |- F/_ w C
3		nfcv 2319	. 2 |- F/_ x D
4		nfcv 2319	. 2 |- F/_ y D
5		cbvmpov.1	. . 3 |- ( x = z -> C = E )
6		cbvmpov.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2230	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpo 5950	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1353   e. cmpo 5873

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4064  df-oprab 5875  df-mpo 5876

This theorem is referenced by:  frec2uzrdg  10403  frecuzrdgsuc  10408  iseqvalcbv  10451  resqrexlemfp1  11010  resqrex  11027  sqne2sq  12168  ennnfonelemnn0  12414  nninfdc  12445  txbas  13620  xmetxp  13869