Theorem mpt2eq123i 5665

Description: An equality inference for the maps to notation. (Contributed by set.mm contributors, 15-Jul-2013.)

Hypotheses

Ref	Expression
mpt2eq123i.1	⊢ A = D
mpt2eq123i.2	⊢ B = E
mpt2eq123i.3	⊢ C = F

Assertion

Ref	Expression
mpt2eq123i	⊢ (x ∈ A, y ∈ B ↦ C) = (x ∈ D, y ∈ E ↦ F)

Proof of Theorem mpt2eq123i

Step	Hyp	Ref	Expression
1		mpt2eq123i.1	. . . 4 ⊢ A = D
2	1	a1i 10	. . 3 ⊢ ( ⊤ → A = D)
3		mpt2eq123i.2	. . . 4 ⊢ B = E
4	3	a1i 10	. . 3 ⊢ ( ⊤ → B = E)
5		mpt2eq123i.3	. . . 4 ⊢ C = F
6	5	a1i 10	. . 3 ⊢ ( ⊤ → C = F)
7	2, 4, 6	mpt2eq123dv 5664	. 2 ⊢ ( ⊤ → (x ∈ A, y ∈ B ↦ C) = (x ∈ D, y ∈ E ↦ F))
8	7	trud 1323	1 ⊢ (x ∈ A, y ∈ B ↦ C) = (x ∈ D, y ∈ E ↦ F)

Syntax hints:   ⊤ wtru 1316   = wceq 1642   ↦ 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: (None)