Theorem mpoeq3ia 5983

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 1017	. . 3 |- ( ( T. /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 5982	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
4	3	mptru 1373	1 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   T. wtru 1365   e. wcel 2164   e. cmpo 5920

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 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  mpodifsnif  6011  mposnif  6012  oprab2co  6271  genpdf  7568  dfioo2  10040  iseqvalcbv  10530  elovmpowrd  10955  dfrhm2  13650  cnfldsub  14063  divcnap  14723