Theorem cbvmpt2 6734

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

Hypotheses

Ref	Expression
cbvmpt2.1	|- F/_ z C
cbvmpt2.2	|- F/_ w C
cbvmpt2.3	|- F/_ x D
cbvmpt2.4	|- F/_ y D
cbvmpt2.5	|- ( ( x = z /\ y = w ) -> C = D )

Assertion

Ref	Expression
cbvmpt2	|- ( 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 cbvmpt2

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ z B
2		nfcv 2764	. 2 |- F/_ x B
3		cbvmpt2.1	. 2 |- F/_ z C
4		cbvmpt2.2	. 2 |- F/_ w C
5		cbvmpt2.3	. 2 |- F/_ x D
6		cbvmpt2.4	. 2 |- F/_ y D
7		eqidd 2623	. 2 |- ( x = z -> B = B )
8		cbvmpt2.5	. 2 |- ( ( x = z /\ y = w ) -> C = D )
9	1, 2, 3, 4, 5, 6, 7, 8	cbvmpt2x 6733	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/_wnfc 2751   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  cbvmpt2v  6735  el2mpt2csbcl  7250  fnmpt2ovd  7252  fmpt2co  7260  mpt2curryd  7395  fvmpt2curryd  7397  xpf1o  8122  cnfcomlem  8596  fseqenlem1  8847  relexpsucnnr  13765  gsumdixp  18609  evlslem4  19508  madugsum  20449  cnmpt2t  21476  cnmptk2  21489  fmucnd  22096  fsum2cn  22674  fmuldfeqlem1  39814  smflim  40985