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Theorem mptv 5265
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 5233 . 2 (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
2 vex 3465 . . . 4 𝑥 ∈ V
32biantrur 529 . . 3 (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵))
43opabbii 5216 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)}
51, 4eqtr4i 2756 1 (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  {copab 5211  cmpt 5232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-opab 5212  df-mpt 5233
This theorem is referenced by:  df1st2  8103  df2nd2  8104  fsplit  8122  rankf  9819
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