Theorem mptv 5719

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

Assertion

Ref	Expression
mptv	⊢ (x ∈ V ↦ B) = {⟨x, y⟩ ∣ y = B}

Distinct variable groups:   x,y   y,B

Allowed substitution hint:   B(x)

Proof of Theorem mptv

Step	Hyp	Ref	Expression
1		df-mpt 5653	. 2 ⊢ (x ∈ V ↦ B) = {⟨x, y⟩ ∣ (x ∈ V ∧ y = B)}
2		vex 2863	. . . 4 ⊢ x ∈ V
3	2	biantrur 492	. . 3 ⊢ (y = B ↔ (x ∈ V ∧ y = B))
4	3	opabbii 4627	. 2 ⊢ {⟨x, y⟩ ∣ y = B} = {⟨x, y⟩ ∣ (x ∈ V ∧ y = B)}
5	1, 4	eqtr4i 2376	1 ⊢ (x ∈ V ↦ B) = {⟨x, y⟩ ∣ y = B}

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {copab 4623   ↦ cmpt 5652

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-opab 4624  df-mpt 5653

This theorem is referenced by:  csucex  6260  brcsuc  6261  frecsuc  6323