Theorem mpteq2ia 4062

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 2164	. . 3 |- A = A
2	1	ax-gen 1436	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2517	. 2 |- A. x e. A B = C
5		mpteq12f 4056	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
6	2, 4, 5	mp2an 423	1 |- ( x e. A |-> B ) = ( x e. A |-> C )

Syntax hints:   -> wi 4   A.wal 1340   = wceq 1342   e. wcel 2135   A.wral 2442   |-> cmpt 4037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-ral 2447  df-opab 4038  df-mpt 4039

This theorem is referenced by:  mpteq2i  4063  feqresmpt  5534  elfvmptrab  5575  fmptap  5669  offres  6095  cnrecnv  10838  ege2le3  11598  eirraplem  11703  cnmpt1st  12829  cnmpt2nd  12830  expcncf  13133  dvexp  13216  dveflem  13228  dvef  13229