Theorem mptv 5218

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 5194	. 2 ⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
2		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 539	. . 3 ⊢ (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵))
4	3	opabbii 5179	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
5	1, 4	eqtr4i 2789	1 ⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {copab 5174   ↦ cmpt 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-opab 5175  df-mpt 5194

This theorem is referenced by:  df1st2  8094  df2nd2  8095  fsplit  8113  rankf  9767  dfsucmap3  39093  dfsucmap4  39095