Theorem mpteq2ia 5659

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

Hypothesis

Ref	Expression
mpteq2ia.1	⊢ (x ∈ A → B = C)

Assertion

Ref	Expression
mpteq2ia	⊢ (x ∈ A ↦ B) = (x ∈ A ↦ C)

Proof of Theorem mpteq2ia

Step	Hyp	Ref	Expression
1		eqid 2353	. . 3 ⊢ A = A
2	1	ax-gen 1546	. 2 ⊢ ∀x A = A
3		mpteq2ia.1	. . 3 ⊢ (x ∈ A → B = C)
4	3	rgen 2679	. 2 ⊢ ∀x ∈ A B = C
5		mpteq12f 5655	. 2 ⊢ ((∀x A = A ∧ ∀x ∈ A B = C) → (x ∈ A ↦ B) = (x ∈ A ↦ C))
6	2, 4, 5	mp2an 653	1 ⊢ (x ∈ A ↦ B) = (x ∈ A ↦ C)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ↦ cmpt 5651

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 2619  df-opab 4623  df-mpt 5652

This theorem is referenced by:  mpteq2i  5660