Theorem mpteq12f 5656

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.

Step	Hyp	Ref	Expression
1		nfa1 1788	. . . 4 ⊢ Ⅎx∀x A = C
2		nfra1 2665	. . . 4 ⊢ Ⅎx∀x ∈ A B = D
3	1, 2	nfan 1824	. . 3 ⊢ Ⅎx(∀x A = C ∧ ∀x ∈ A B = D)
4		nfv 1619	. . 3 ⊢ Ⅎy(∀x A = C ∧ ∀x ∈ A B = D)
5		rsp 2675	. . . . . . 7 ⊢ (∀x ∈ A B = D → (x ∈ A → B = D))
6	5	imp 418	. . . . . 6 ⊢ ((∀x ∈ A B = D ∧ x ∈ A) → B = D)
7	6	eqeq2d 2364	. . . . 5 ⊢ ((∀x ∈ A B = D ∧ x ∈ A) → (y = B ↔ y = D))
8	7	pm5.32da 622	. . . 4 ⊢ (∀x ∈ A B = D → ((x ∈ A ∧ y = B) ↔ (x ∈ A ∧ y = D)))
9		sp 1747	. . . . . 6 ⊢ (∀x A = C → A = C)
10	9	eleq2d 2420	. . . . 5 ⊢ (∀x A = C → (x ∈ A ↔ x ∈ C))
11	10	anbi1d 685	. . . 4 ⊢ (∀x A = C → ((x ∈ A ∧ y = D) ↔ (x ∈ C ∧ y = D)))
12	8, 11	sylan9bbr 681	. . 3 ⊢ ((∀x A = C ∧ ∀x ∈ A B = D) → ((x ∈ A ∧ y = B) ↔ (x ∈ C ∧ y = D)))
13	3, 4, 12	opabbid 4625	. 2 ⊢ ((∀x A = C ∧ ∀x ∈ A B = D) → {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)} = {⟨x, y⟩ ∣ (x ∈ C ∧ y = D)})
14		df-mpt 5653	. 2 ⊢ (x ∈ A ↦ B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
15		df-mpt 5653	. 2 ⊢ (x ∈ C ↦ D) = {⟨x, y⟩ ∣ (x ∈ C ∧ y = D)}
16	13, 14, 15	3eqtr4g 2410	1 ⊢ ((∀x A = C ∧ ∀x ∈ A B = D) → (x ∈ A ↦ B) = (x ∈ C ↦ D))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {copab 4623   ↦ cmpt 5652

This proof depends on 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 proof 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 used by:  mpteq12dv  5657  mpteq12  5658  mpteq2ia  5660  mpteq2da  5667