Theorem mpteq2ia 4198

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpteq2ia.1	|- ( x e. A -> B = C )

Assertion

Ref	Expression
mpteq2ia	|- ( x e. A |-> B ) = ( x e. A |-> C )

Proof of Theorem mpteq2ia

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 |- A = A
2	1	ax-gen 1498	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2597	. 2 |- A. x e. A B = C
5		mpteq12f 4192	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
6	2, 4, 5	mp2an 426	1 |- ( x e. A |-> B ) = ( x e. A |-> C )

Syntax hints:   -> wi 4   A.wal 1396   = wceq 1398   e. wcel 2205   A.wral 2522   |-> cmpt 4173

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-ral 2527  df-opab 4174  df-mpt 4175

This theorem is referenced by:  mpteq2i  4199  feqresmpt  5733  elfvmptrab  5775  fmptap  5876  offres  6330  cnrecnv  11599  ege2le3  12361  eirraplem  12467  cnmpt1st  15170  cnmpt2nd  15171  expcn  15451  expcncf  15491  dvexp  15593  dveflem  15608  dvef  15609  elply2  15617  plyid  15628