Theorem mptv 5258

Description: Function with universal domain in maps-to notation. (Contributed by NM, 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 5226	. 2 ⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
2		vex 3484	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 530	. . 3 ⊢ (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵))
4	3	opabbii 5210	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
5	1, 4	eqtr4i 2768	1 ⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {copab 5205   ↦ cmpt 5225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-opab 5206  df-mpt 5226

This theorem is referenced by:  df1st2  8123  df2nd2  8124  fsplit  8142  rankf  9834