Theorem cbvmpo 6024

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 2348	. 2 |- F/_ z B
2		nfcv 2348	. 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 2206	. 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 6023	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   F/_wnfc 2335   e. cmpo 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-oprab 5948  df-mpo 5949

This theorem is referenced by:  cbvmpov  6025  fvmpopr2d  6082  fnmpoovd  6301  fmpoco  6302  xpf1o  6941  cnmpt2t  14765