Theorem mpoeq123dva 5840

Description: An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
mpoeq123dv.1	|- ( ph -> A = D )
mpoeq123dva.2	|- ( ( ph /\ x e. A ) -> B = E )
mpoeq123dva.3	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )

Assertion

Ref	Expression
mpoeq123dva	|- ( 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 mpoeq123dva

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpoeq123dva.3	. . . . . 6 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )
2	1	eqeq2d 2152	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z = C <-> z = F ) )
3	2	pm5.32da 448	. . . 4 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ z = F ) ) )
4		mpoeq123dva.2	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B = E )
5	4	eleq2d 2210	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( y e. B <-> y e. E ) )
6	5	pm5.32da 448	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. B ) <-> ( x e. A /\ y e. E ) ) )
7		mpoeq123dv.1	. . . . . . . 8 |- ( ph -> A = D )
8	7	eleq2d 2210	. . . . . . 7 |- ( ph -> ( x e. A <-> x e. D ) )
9	8	anbi1d 461	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. E ) <-> ( x e. D /\ y e. E ) ) )
10	6, 9	bitrd 187	. . . . 5 |- ( ph -> ( ( x e. A /\ y e. B ) <-> ( x e. D /\ y e. E ) ) )
11	10	anbi1d 461	. . . 4 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = F ) <-> ( ( x e. D /\ y e. E ) /\ z = F ) ) )
12	3, 11	bitrd 187	. . 3 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. D /\ y e. E ) /\ z = F ) ) )
13	12	oprabbidv 5833	. 2 |- ( ph -> { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } = { <. <. x , y >. , z >. | ( ( x e. D /\ y e. E ) /\ z = F ) } )
14		df-mpo 5787	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
15		df-mpo 5787	. 2 |- ( x e. D , y e. E |-> F ) = { <. <. x , y >. , z >. | ( ( x e. D /\ y e. E ) /\ z = F ) }
16	13, 14, 15	3eqtr4g 2198	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {coprab 5783   e. cmpo 5784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-oprab 5786  df-mpo 5787

This theorem is referenced by:  mpoeq123dv  5841