Theorem mpteq12f 5656

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 1788	. . . 4 |- F/xA.x A = C
2		nfra1 2665	. . . 4 |- F/xA.x e. A B = D
3	1, 2	nfan 1824	. . 3 |- F/x(A.x A = C /\ A.x e. A B = D)
4		nfv 1619	. . 3 |- F/y(A.x A = C /\ A.x e. A B = D)
5		rsp 2675	. . . . . . 7 |- (A.x e. A B = D -> (x e. A -> B = D))
6	5	imp 418	. . . . . 6 |- ((A.x e. A B = D /\ x e. A) -> B = D)
7	6	eqeq2d 2364	. . . . 5 |- ((A.x e. A B = D /\ x e. A) -> (y = B <-> y = D))
8	7	pm5.32da 622	. . . 4 |- (A.x e. A B = D -> ((x e. A /\ y = B) <-> (x e. A /\ y = D)))
9		sp 1747	. . . . . 6 |- (A.x A = C -> A = C)
10	9	eleq2d 2420	. . . . 5 |- (A.x A = C -> (x e. A <-> x e. C))
11	10	anbi1d 685	. . . 4 |- (A.x A = C -> ((x e. A /\ y = D) <-> (x e. C /\ y = D)))
12	8, 11	sylan9bbr 681	. . 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 4625	. 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 5653	. 2 |- (x e. A |-> B) = {<.x, y>. | (x e. A /\ y = B)}
15		df-mpt 5653	. 2 |- (x e. C |-> D) = {<.x, y>. | (x e. C /\ y = D)}
16	13, 14, 15	3eqtr4g 2410	1 |- ((A.x A = C /\ A.x e. A B = D) -> (x e. A |-> B) = (x e. C |-> D))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2615  {copab 4623   |-> cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-opab 4624  df-mpt 5653

This theorem is referenced by:  mpteq12dv  5657  mpteq12  5658  mpteq2ia  5660  mpteq2da  5667