Theorem mpoeq3ia 6143

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 1046	. . 3 |- ( ( T. /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 6142	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
4	3	mptru 1411	1 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   T. wtru 1403   e. wcel 2209   e. cmpo 6077

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-oprab 6079  df-mpo 6080

This theorem is referenced by:  mpodifsnif  6171  mposnif  6172  oprab2co  6444  genpdf  7865  dfioo2  10355  iseqvalcbv  10874  elovmpowrd  11324  dfrhm2  14434  cnfldsub  14884  divcnap  15589