Theorem mpt2eq123dva 6716

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

Hypotheses

Ref	Expression
mpt2eq123dv.1	|- ( ph -> A = D )
mpt2eq123dva.2	|- ( ( ph /\ x e. A ) -> B = E )
mpt2eq123dva.3	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )

Assertion

Ref	Expression
mpt2eq123dva	|- ( 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 mpt2eq123dva

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2eq123dva.3	. . . . . 6 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )
2	1	eqeq2d 2632	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z = C <-> z = F ) )
3	2	pm5.32da 673	. . . 4 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ z = F ) ) )
4		mpt2eq123dva.2	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B = E )
5	4	eleq2d 2687	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( y e. B <-> y e. E ) )
6	5	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. B ) <-> ( x e. A /\ y e. E ) ) )
7		mpt2eq123dv.1	. . . . . . . 8 |- ( ph -> A = D )
8	7	eleq2d 2687	. . . . . . 7 |- ( ph -> ( x e. A <-> x e. D ) )
9	8	anbi1d 741	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. E ) <-> ( x e. D /\ y e. E ) ) )
10	6, 9	bitrd 268	. . . . 5 |- ( ph -> ( ( x e. A /\ y e. B ) <-> ( x e. D /\ y e. E ) ) )
11	10	anbi1d 741	. . . 4 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = F ) <-> ( ( x e. D /\ y e. E ) /\ z = F ) ) )
12	3, 11	bitrd 268	. . 3 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. D /\ y e. E ) /\ z = F ) ) )
13	12	oprabbidv 6709	. 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-mpt2 6655	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
15		df-mpt2 6655	. 2 |- ( x e. D , y e. E |-> F ) = { <. <. x , y >. , z >. | ( ( x e. D /\ y e. E ) /\ z = F ) }
16	13, 14, 15	3eqtr4g 2681	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {coprab 6651   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  mpt2eq123dv  6717  natpropd  16636  fucpropd  16637  curfpropd  16873  hofpropd  16907  istrkgl  25357  eengv  25859  elntg  25864  submat1n  29871  rrxdsfi  40505  rrxtopnfi  40506  rngcifuestrc  41997  funcrngcsetc  41998  funcrngcsetcALT  41999  funcringcsetc  42035