Theorem mpteq12i 4148

Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12i.1	|- A = C
mpteq12i.2	|- B = D

Assertion

Ref	Expression
mpteq12i	|- ( x e. A |-> B ) = ( x e. C |-> D )

Proof of Theorem mpteq12i

Step	Hyp	Ref	Expression
1		mpteq12i.1	. . . 4 |- A = C
2	1	a1i 9	. . 3 |- ( T. -> A = C )
3		mpteq12i.2	. . . 4 |- B = D
4	3	a1i 9	. . 3 |- ( T. -> B = D )
5	2, 4	mpteq12dv 4142	. 2 |- ( T. -> ( x e. A |-> B ) = ( x e. C |-> D ) )
6	5	mptru 1382	1 |- ( x e. A |-> B ) = ( x e. C |-> D )

Syntax hints:   = wceq 1373   T. wtru 1374   |-> 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:  offres  6243