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Theorem mptv 3844
 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 3811 . 2 (x V ↦ B) = {⟨x, y⟩ ∣ (x V y = B)}
2 vex 2554 . . . 4 x V
32biantrur 287 . . 3 (y = B ↔ (x V y = B))
43opabbii 3815 . 2 {⟨x, y⟩ ∣ y = B} = {⟨x, y⟩ ∣ (x V y = B)}
51, 4eqtr4i 2060 1 (x V ↦ B) = {⟨x, y⟩ ∣ y = B}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {copab 3808   ↦ cmpt 3809 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-opab 3810  df-mpt 3811 This theorem is referenced by:  df1st2  5782  df2nd2  5783
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