 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptv Structured version   Visualization version   GIF version

Theorem mptv 5029
 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
StepHypRef Expression
1 df-mpt 5009 . 2 (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
2 vex 3418 . . . 4 𝑥 ∈ V
32biantrur 523 . . 3 (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵))
43opabbii 4996 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
51, 4eqtr4i 2805 1 (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3415  {copab 4991   ↦ cmpt 5008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-v 3417  df-opab 4992  df-mpt 5009 This theorem is referenced by:  df1st2  7601  df2nd2  7602  fsplit  7620  rankf  9017
 Copyright terms: Public domain W3C validator