Theorem cbvmpo 6083

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 2372	. 2 |- F/_ z B
2		nfcv 2372	. 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 2230	. 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 6082	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   F/_wnfc 2359   e. cmpo 6003

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-oprab 6005  df-mpo 6006

This theorem is referenced by:  cbvmpov  6084  fvmpopr2d  6141  fnmpoovd  6361  fmpoco  6362  xpf1o  7005  cnmpt2t  14967