Theorem mpoeq123dv 6082

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 6081	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   e. cmpo 6019

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-oprab 6021  df-mpo 6022

This theorem is referenced by:  mpoeq123i  6083  prdsex  13351  prdsval  13355  plusffvalg  13444  grpsubfvalg  13627  grpsubpropdg  13686  mulgfvalg  13707  mulgpropdg  13750  dvrfvald  14146  scaffvalg  14319  psrval  14679  blfvalps  15108  clwwlknonmpo  16278