Theorem mpt2eq123i 5665

Description: An equality inference for the maps to notation. (Contributed by set.mm contributors, 15-Jul-2013.)

Hypotheses

Ref	Expression
mpt2eq123i.1	|- A = D
mpt2eq123i.2	|- B = E
mpt2eq123i.3	|- C = F

Assertion

Ref	Expression
mpt2eq123i	|- (x e. A, y e. B |-> C) = (x e. D, y e. E |-> F)

Proof of Theorem mpt2eq123i

Step	Hyp	Ref	Expression
1		mpt2eq123i.1	. . . 4 |- A = D
2	1	a1i 10	. . 3 |- ( T. -> A = D)
3		mpt2eq123i.2	. . . 4 |- B = E
4	3	a1i 10	. . 3 |- ( T. -> B = E)
5		mpt2eq123i.3	. . . 4 |- C = F
6	5	a1i 10	. . 3 |- ( T. -> C = F)
7	2, 4, 6	mpt2eq123dv 5664	. 2 |- ( T. -> (x e. A, y e. B |-> C) = (x e. D, y e. E |-> F))
8	7	trud 1323	1 |- (x e. A, y e. B |-> C) = (x e. D, y e. E |-> F)

Syntax hints:   T. wtru 1316   = wceq 1642   |-> cmpt2 5654

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-oprab 5529  df-mpt2 5655

This theorem is referenced by: (None)