Theorem mpteq1 5658

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

Assertion

Ref	Expression
mpteq1	⊢ (A = B → (x ∈ A ↦ C) = (x ∈ B ↦ C))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem mpteq1

Step	Hyp	Ref	Expression
1		eqidd 2354	. . 3 ⊢ (x ∈ A → C = C)
2	1	rgen 2679	. 2 ⊢ ∀x ∈ A C = C
3		mpteq12 5657	. 2 ⊢ ((A = B ∧ ∀x ∈ A C = C) → (x ∈ A ↦ C) = (x ∈ B ↦ C))
4	2, 3	mpan2 652	1 ⊢ (A = B → (x ∈ A ↦ C) = (x ∈ B ↦ C))

Syntax hints:   → wi 4   = 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:  mpt2mpt  5709  elscan  6330