Theorem mpteq12i 5666

Description: An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.)

Hypotheses

Ref	Expression
mpteq12i.1	⊢ A = C
mpteq12i.2	⊢ B = D

Assertion

Ref	Expression
mpteq12i	⊢ (x ∈ A ↦ B) = (x ∈ C ↦ D)

Proof of Theorem mpteq12i

Step	Hyp	Ref	Expression
1		mpteq12i.1	. . . 4 ⊢ A = C
2	1	a1i 10	. . 3 ⊢ ( ⊤ → A = C)
3		mpteq12i.2	. . . 4 ⊢ B = D
4	3	a1i 10	. . 3 ⊢ ( ⊤ → B = D)
5	2, 4	mpteq12dv 5657	. 2 ⊢ ( ⊤ → (x ∈ A ↦ B) = (x ∈ C ↦ D))
6	5	trud 1323	1 ⊢ (x ∈ A ↦ B) = (x ∈ C ↦ D)

Syntax hints:   ⊤ wtru 1316   = wceq 1642   ↦ 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: (None)