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	⊢ (φ → A = D)
mpt2eq123dv.2	⊢ (φ → B = E)
mpt2eq123dv.3	⊢ (φ → C = F)

Assertion

Ref	Expression
mpt2eq123dv	⊢ (φ → (x ∈ A, y ∈ B ↦ C) = (x ∈ D, y ∈ E ↦ F))

Distinct variable groups:   φ,x   φ,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 ⊢ (φ → A = D)
2	1	eleq2d 2420	. . . . 5 ⊢ (φ → (x ∈ A ↔ x ∈ D))
3		mpt2eq123dv.2	. . . . . 6 ⊢ (φ → B = E)
4	3	eleq2d 2420	. . . . 5 ⊢ (φ → (y ∈ B ↔ y ∈ E))
5	2, 4	anbi12d 691	. . . 4 ⊢ (φ → ((x ∈ A ∧ y ∈ B) ↔ (x ∈ D ∧ y ∈ E)))
6		mpt2eq123dv.3	. . . . 5 ⊢ (φ → C = F)
7	6	eqeq2d 2364	. . . 4 ⊢ (φ → (z = C ↔ z = F))
8	5, 7	anbi12d 691	. . 3 ⊢ (φ → (((x ∈ A ∧ y ∈ B) ∧ z = C) ↔ ((x ∈ D ∧ y ∈ E) ∧ z = F)))
9	8	oprabbidv 5565	. 2 ⊢ (φ → {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ D ∧ y ∈ E) ∧ z = F)})
10		df-mpt2 5655	. 2 ⊢ (x ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
11		df-mpt2 5655	. 2 ⊢ (x ∈ D, y ∈ E ↦ F) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ D ∧ y ∈ E) ∧ z = F)}
12	9, 10, 11	3eqtr4g 2410	1 ⊢ (φ → (x ∈ A, y ∈ B ↦ C) = (x ∈ D, y ∈ E ↦ F))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {coprab 5528   ↦ cmpt2 5654

This proof depends on 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 proof 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 used by:  mpt2eq123i  5665