Theorem mpteq12f 4109

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 1552	. . . 4 |- F/ x A. x A = C
2		nfra1 2525	. . . 4 |- F/ x A. x e. A B = D
3	1, 2	nfan 1576	. . 3 |- F/ x ( A. x A = C /\ A. x e. A B = D )
4		nfv 1539	. . 3 |- F/ y ( A. x A = C /\ A. x e. A B = D )
5		rsp 2541	. . . . . . 7 |- ( A. x e. A B = D -> ( x e. A -> B = D ) )
6	5	imp 124	. . . . . 6 |- ( ( A. x e. A B = D /\ x e. A ) -> B = D )
7	6	eqeq2d 2205	. . . . 5 |- ( ( A. x e. A B = D /\ x e. A ) -> ( y = B <-> y = D ) )
8	7	pm5.32da 452	. . . 4 |- ( A. x e. A B = D -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
9		sp 1522	. . . . . 6 |- ( A. x A = C -> A = C )
10	9	eleq2d 2263	. . . . 5 |- ( A. x A = C -> ( x e. A <-> x e. C ) )
11	10	anbi1d 465	. . . 4 |- ( A. x A = C -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
12	8, 11	sylan9bbr 463	. . 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 4094	. 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 4092	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
15		df-mpt 4092	. 2 |- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
16	13, 14, 15	3eqtr4g 2251	1 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   {copab 4089   |-> cmpt 4090

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-opab 4091  df-mpt 4092

This theorem is referenced by:  mpteq12dva  4110  mpteq12  4112  mpteq2ia  4115  mpteq2da  4118