Theorem mpoeq3ia 5918

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpoeq3ia.1	|- ( ( x e. A /\ y e. B ) -> C = D )

Assertion

Ref	Expression
mpoeq3ia	|- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Proof of Theorem mpoeq3ia

Step	Hyp	Ref	Expression
1		mpoeq3ia.1	. . . 4 |- ( ( x e. A /\ y e. B ) -> C = D )
2	1	3adant1 1010	. . 3 |- ( ( T. /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 5917	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
4	3	mptru 1357	1 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   T. wtru 1349   e. wcel 2141   e. cmpo 5855

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  mpodifsnif  5946  mposnif  5947  oprab2co  6197  genpdf  7470  dfioo2  9931  iseqvalcbv  10413  divcnap  13349