Theorem mpteq2ia 4146

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 2207	. . 3 |- A = A
2	1	ax-gen 1473	. 2 |- A. x A = A
3		mpteq2ia.1	. . 3 |- ( x e. A -> B = C )
4	3	rgen 2561	. 2 |- A. x e. A B = C
5		mpteq12f 4140	. 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 1371   = wceq 1373   e. wcel 2178   A.wral 2486   |-> cmpt 4121

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-opab 4122  df-mpt 4123

This theorem is referenced by:  mpteq2i  4147  feqresmpt  5656  elfvmptrab  5698  fmptap  5797  offres  6243  cnrecnv  11336  ege2le3  12097  eirraplem  12203  cnmpt1st  14875  cnmpt2nd  14876  expcn  15156  expcncf  15196  dvexp  15298  dveflem  15313  dvef  15314  elply2  15322  plyid  15333