Theorem mpt2eq123i 5750

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 5749	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
8	7	mptru 1305	1 |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Syntax hints:   = wceq 1296   T. wtru 1297   |-> cmpt2 5692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-oprab 5694  df-mpt2 5695

This theorem is referenced by:  ofmres  5945