Theorem mpteq1 5659

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

Assertion

Ref	Expression
mpteq1	|- (A = B -> (x e. A |-> C) = (x e. 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 e. A -> C = C)
2	1	rgen 2680	. 2 |- A.x e. A C = C
3		mpteq12 5658	. 2 |- ((A = B /\ A.x e. A C = C) -> (x e. A |-> C) = (x e. B |-> C))
4	2, 3	mpan2 652	1 |- (A = B -> (x e. A |-> C) = (x e. B |-> C))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  A.wral 2615   |-> 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:  mpt2mpt  5710  elscan  6331