Theorem cbvmpo 6001

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 2339	. 2 |- F/_ z B
2		nfcv 2339	. 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 2197	. 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 6000	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   F/_wnfc 2326   e. cmpo 5924

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-oprab 5926  df-mpo 5927

This theorem is referenced by:  cbvmpov  6002  fvmpopr2d  6059  fnmpoovd  6273  fmpoco  6274  xpf1o  6905  cnmpt2t  14529