Theorem cbvmpo 6110

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)

Hypotheses

Ref	Expression
cbvmpo.1	|- F/_ z C
cbvmpo.2	|- F/_ w C
cbvmpo.3	|- F/_ x D
cbvmpo.4	|- F/_ y D
cbvmpo.5	|- ( ( x = z /\ y = w ) -> C = D )

Assertion

Ref	Expression
cbvmpo	|- ( 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

Allowed substitution hints:   C( x, y, z, w)   D( x, y, z, w)

Proof of Theorem cbvmpo

Step	Hyp	Ref	Expression
1		nfcv 2375	. 2 |- F/_ z B
2		nfcv 2375	. 2 |- F/_ x B
3		cbvmpo.1	. 2 |- F/_ z C
4		cbvmpo.2	. 2 |- F/_ w C
5		cbvmpo.3	. 2 |- F/_ x D
6		cbvmpo.4	. 2 |- F/_ y D
7		eqidd 2232	. 2 |- ( x = z -> B = B )
8		cbvmpo.5	. 2 |- ( ( x = z /\ y = w ) -> C = D )
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpox 6109	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   F/_wnfc 2362   e. cmpo 6030

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-oprab 6032  df-mpo 6033

This theorem is referenced by:  cbvmpov  6111  fvmpopr2d  6168  fnmpoovd  6389  fmpoco  6390  xpf1o  7073  cnmpt2t  15104