Theorem cbvmpov 6168

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4226, 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 2392	. 2 |- F/_ z C
2		nfcv 2392	. 2 |- F/_ w C
3		nfcv 2392	. 2 |- F/_ x D
4		nfcv 2392	. 2 |- F/_ y D
5		cbvmpov.1	. . 3 |- ( x = z -> C = E )
6		cbvmpov.2	. . 3 |- ( y = w -> E = D )
7	5, 6	sylan9eq 2291	. 2 |- ( ( x = z /\ y = w ) -> C = D )
8	1, 2, 3, 4, 7	cbvmpo 6167	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. cmpo 6087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-oprab 6089  df-mpo 6090

This theorem is used by:  frec2uzrdg  10846  frecuzrdgsuc  10851  iseqvalcbv  10896  resqrexlemfp1  11775  resqrex  11792  sqne2sq  12955  ennnfonelemnn0  13313  nninfdc  13344  txbas  15359  xmetxp  15608  mpomulcn  15667  depindlem1  16747