Theorem cbvmpov 6101

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4184, 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 2374	. 2 |- F/_ z C
2		nfcv 2374	. 2 |- F/_ w C
3		nfcv 2374	. 2 |- F/_ x D
4		nfcv 2374	. 2 |- F/_ y D
5		cbvmpov.1	. . 3 |- ( x = z -> C = E )
6		cbvmpov.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2284	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpo 6100	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1397   e. cmpo 6020

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-oprab 6022  df-mpo 6023

This theorem is referenced by:  frec2uzrdg  10672  frecuzrdgsuc  10677  iseqvalcbv  10722  resqrexlemfp1  11587  resqrex  11604  sqne2sq  12767  ennnfonelemnn0  13061  nninfdc  13092  txbas  15001  xmetxp  15250  mpomulcn  15309  depindlem1  16376