Theorem mpt2v 5720

Description: Operation with universal domain in maps-to notation. (Contributed by set.mm contributors, 16-Aug-2013.)

Assertion

Ref	Expression
mpt2v	⊢ (x ∈ V, y ∈ V ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ z = C}

Distinct variable groups:   x,z   y,z   z,C

Allowed substitution hints:   C(x,y)

Proof of Theorem mpt2v

Step	Hyp	Ref	Expression
1		df-mpt2 5655	. 2 ⊢ (x ∈ V, y ∈ V ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = C)}
2		vex 2863	. . . . 5 ⊢ x ∈ V
3		vex 2863	. . . . 5 ⊢ y ∈ V
4	2, 3	pm3.2i 441	. . . 4 ⊢ (x ∈ V ∧ y ∈ V)
5	4	biantrur 492	. . 3 ⊢ (z = C ↔ ((x ∈ V ∧ y ∈ V) ∧ z = C))
6	5	oprabbii 5566	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ z = C} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = C)}
7	1, 6	eqtr4i 2376	1 ⊢ (x ∈ V, y ∈ V ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ z = C}

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {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-v 2862  df-oprab 5529  df-mpt2 5655

This theorem is referenced by:  dfswap4  5730