Theorem mpoeq123i 5834

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

Hypotheses

Ref	Expression
mpoeq123i.1	|- A = D
mpoeq123i.2	|- B = E
mpoeq123i.3	|- C = F

Assertion

Ref	Expression
mpoeq123i	|- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Proof of Theorem mpoeq123i

Step	Hyp	Ref	Expression
1		mpoeq123i.1	. . . 4 |- A = D
2	1	a1i 9	. . 3 |- ( T. -> A = D )
3		mpoeq123i.2	. . . 4 |- B = E
4	3	a1i 9	. . 3 |- ( T. -> B = E )
5		mpoeq123i.3	. . . 4 |- C = F
6	5	a1i 9	. . 3 |- ( T. -> C = F )
7	2, 4, 6	mpoeq123dv 5833	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
8	7	mptru 1340	1 |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Syntax hints:   = wceq 1331   T. wtru 1332   e. cmpo 5776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  ofmres  6034