Theorem mpoeq123i 5954

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 5953	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
8	7	mptru 1373	1 |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Syntax hints:   = wceq 1364   T. wtru 1365   e. cmpo 5893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-oprab 5895  df-mpo 5896

This theorem is referenced by:  ofmres  6155