Theorem mpteq2ia 4141

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 2206	. . 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 2560	. 2 |- A. x e. A B = C
5		mpteq12f 4135	. 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 2177   A.wral 2485   |-> cmpt 4116

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 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ral 2490  df-opab 4117  df-mpt 4118

This theorem is referenced by:  mpteq2i  4142  feqresmpt  5651  elfvmptrab  5693  fmptap  5792  offres  6238  cnrecnv  11306  ege2le3  12067  eirraplem  12173  cnmpt1st  14845  cnmpt2nd  14846  expcn  15126  expcncf  15166  dvexp  15268  dveflem  15283  dvef  15284  elply2  15292  plyid  15303