Theorem mpoeq3ia 5836

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 999	. . 3 |- ( ( T. /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 5835	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
4	3	mptru 1340	1 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   T. wtru 1332   e. wcel 1480   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-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  mpodifsnif  5864  mposnif  5865  oprab2co  6115  genpdf  7316  dfioo2  9757  iseqvalcbv  10230  divcnap  12724