Theorem mpt2eq123dv 5664

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

Hypotheses

Ref	Expression
mpt2eq123dv.1	|- (ph -> A = D)
mpt2eq123dv.2	|- (ph -> B = E)
mpt2eq123dv.3	|- (ph -> C = F)

Assertion

Ref	Expression
mpt2eq123dv	|- (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 mpt2eq123dv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2eq123dv.1	. . . . . 6 |- (ph -> A = D)
2	1	eleq2d 2420	. . . . 5 |- (ph -> (x e. A <-> x e. D))
3		mpt2eq123dv.2	. . . . . 6 |- (ph -> B = E)
4	3	eleq2d 2420	. . . . 5 |- (ph -> (y e. B <-> y e. E))
5	2, 4	anbi12d 691	. . . 4 |- (ph -> ((x e. A /\ y e. B) <-> (x e. D /\ y e. E)))
6		mpt2eq123dv.3	. . . . 5 |- (ph -> C = F)
7	6	eqeq2d 2364	. . . 4 |- (ph -> (z = C <-> z = F))
8	5, 7	anbi12d 691	. . 3 |- (ph -> (((x e. A /\ y e. B) /\ z = C) <-> ((x e. D /\ y e. E) /\ z = F)))
9	8	oprabbidv 5565	. 2 |- (ph -> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.<.x, y>., z>. | ((x e. D /\ y e. E) /\ z = F)})
10		df-mpt2 5655	. 2 |- (x e. A, y e. B |-> C) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
11		df-mpt2 5655	. 2 |- (x e. D, y e. E |-> F) = {<.<.x, y>., z>. | ((x e. D /\ y e. E) /\ z = F)}
12	9, 10, 11	3eqtr4g 2410	1 |- (ph -> (x e. A, y e. B |-> C) = (x e. D, y e. E |-> F))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  {coprab 5528   |-> cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-oprab 5529  df-mpt2 5655

This theorem is referenced by:  mpt2eq123i  5665