Theorem mpteq12f 4069

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

Assertion

Ref	Expression
mpteq12f	|- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof of Theorem mpteq12f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1534	. . . 4 |- F/ x A. x A = C
2		nfra1 2501	. . . 4 |- F/ x A. x e. A B = D
3	1, 2	nfan 1558	. . 3 |- F/ x ( A. x A = C /\ A. x e. A B = D )
4		nfv 1521	. . 3 |- F/ y ( A. x A = C /\ A. x e. A B = D )
5		rsp 2517	. . . . . . 7 |- ( A. x e. A B = D -> ( x e. A -> B = D ) )
6	5	imp 123	. . . . . 6 |- ( ( A. x e. A B = D /\ x e. A ) -> B = D )
7	6	eqeq2d 2182	. . . . 5 |- ( ( A. x e. A B = D /\ x e. A ) -> ( y = B <-> y = D ) )
8	7	pm5.32da 449	. . . 4 |- ( A. x e. A B = D -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
9		sp 1504	. . . . . 6 |- ( A. x A = C -> A = C )
10	9	eleq2d 2240	. . . . 5 |- ( A. x A = C -> ( x e. A <-> x e. C ) )
11	10	anbi1d 462	. . . 4 |- ( A. x A = C -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
12	8, 11	sylan9bbr 460	. . 3 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
13	3, 4, 12	opabbid 4054	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
14		df-mpt 4052	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
15		df-mpt 4052	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
16	13, 14, 15	3eqtr4g 2228	1 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   {copab 4049   |-> cmpt 4050

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-opab 4051  df-mpt 4052

This theorem is referenced by:  mpteq12dva  4070  mpteq12  4072  mpteq2ia  4075  mpteq2da  4078