Theorem mpt2eq123i 5568

Description: An equality inference for the maps to notation. (Contributed by NM, 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 9	. . 3 |- ( T. -> A = D )
3		mpt2eq123i.2	. . . 4 |- B = E
4	3	a1i 9	. . 3 |- ( T. -> B = E )
5		mpt2eq123i.3	. . . 4 |- C = F
6	5	a1i 9	. . 3 |- ( T. -> C = F )
7	2, 4, 6	mpt2eq123dv 5567	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
8	7	trud 1252	1 |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Syntax hints:   = wceq 1243   T. wtru 1244   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  ofmres  5763