Theorem mpoeq123dv 5980

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 5979	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   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-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  mpoeq123i  5981  prdsex  12880  plusffvalg  12945  grpsubfvalg  13117  grpsubpropdg  13176  mulgfvalg  13191  mulgpropdg  13234  dvrfvald  13629  scaffvalg  13802  psrval  14152  blfvalps  14553