Theorem mpoeq123dv 5940

Description: An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)

Hypotheses

Ref	Expression
mpoeq123dv.1	|- ( ph -> A = D )
mpoeq123dv.2	|- ( ph -> B = E )
mpoeq123dv.3	|- ( ph -> C = F )

Assertion

Ref	Expression
mpoeq123dv	|- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   D( x, y)   E( x, y)   F( x, y)

Proof of Theorem mpoeq123dv

Step	Hyp	Ref	Expression
1		mpoeq123dv.1	. 2 |- ( ph -> A = D )
2		mpoeq123dv.2	. . 3 |- ( ph -> B = E )
3	2	adantr 276	. 2 |- ( ( ph /\ x e. A ) -> B = E )
4		mpoeq123dv.3	. . 3 |- ( ph -> C = F )
5	4	adantr 276	. 2 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )
6	1, 3, 5	mpoeq123dva 5939	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   e. cmpo 5880

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-oprab 5882  df-mpo 5883

This theorem is referenced by:  mpoeq123i  5941  prdsex  12724  plusffvalg  12788  grpsubfvalg  12925  grpsubpropdg  12981  mulgfvalg  12994  mulgpropdg  13035  dvrfvald  13313  scaffvalg  13407  blfvalps  14046