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Theorem mptv 4679
 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 4645 . 2 (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
2 vex 3176 . . . 4 𝑥 ∈ V
32biantrur 526 . . 3 (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵))
43opabbii 4649 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
51, 4eqtr4i 2635 1 (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {copab 4642   ↦ cmpt 4643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-opab 4644  df-mpt 4645 This theorem is referenced by:  df1st2  7150  df2nd2  7151  fsplit  7169  rankf  8540  cnmptid  21274
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