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Theorem mpteq12f 5655
 Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12f ((x A = C x A B = D) → (x A B) = (x C D))

Proof of Theorem mpteq12f
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfa1 1788 . . . 4 xx A = C
2 nfra1 2664 . . . 4 xx A B = D
31, 2nfan 1824 . . 3 x(x A = C x A B = D)
4 nfv 1619 . . 3 y(x A = C x A B = D)
5 rsp 2674 . . . . . . 7 (x A B = D → (x AB = D))
65imp 418 . . . . . 6 ((x A B = D x A) → B = D)
76eqeq2d 2364 . . . . 5 ((x A B = D x A) → (y = By = D))
87pm5.32da 622 . . . 4 (x A B = D → ((x A y = B) ↔ (x A y = D)))
9 sp 1747 . . . . . 6 (x A = CA = C)
109eleq2d 2420 . . . . 5 (x A = C → (x Ax C))
1110anbi1d 685 . . . 4 (x A = C → ((x A y = D) ↔ (x C y = D)))
128, 11sylan9bbr 681 . . 3 ((x A = C x A B = D) → ((x A y = B) ↔ (x C y = D)))
133, 4, 12opabbid 4624 . 2 ((x A = C x A B = D) → {x, y (x A y = B)} = {x, y (x C y = D)})
14 df-mpt 5652 . 2 (x A B) = {x, y (x A y = B)}
15 df-mpt 5652 . 2 (x C D) = {x, y (x C y = D)}
1613, 14, 153eqtr4g 2410 1 ((x A = C x A B = D) → (x A B) = (x C D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {copab 4622   ↦ cmpt 5651 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2619  df-opab 4623  df-mpt 5652 This theorem is referenced by:  mpteq12dv  5656  mpteq12  5657  mpteq2ia  5659  mpteq2da  5666
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