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Theorem ovprc2 7453
Description: The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
ovprc1.1 Rel dom 𝐹
Assertion
Ref Expression
ovprc2 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)

Proof of Theorem ovprc2
StepHypRef Expression
1 simpr 483 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2 ovprc1.1 . . 3 Rel dom 𝐹
32ovprc 7451 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
41, 3nsyl5 159 1 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  c0 4323  dom cdm 5677  Rel wrel 5682  (class class class)co 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7416
This theorem is referenced by:  ressbasssg  17187  ressbasssOLD  17190  ress0  17194  wunress  17201  wunressOLD  17202  0rest  17381  firest  17384  subcmn  19748  dprdval0prc  19915  zrhval  21278  dsmmval2  21512  psrbas  21718  psr1val  21931  vr1val  21937  ply1ascl  22002  evl1fval  22069  restbas  22884  resstopn  22912  deg1fval  25832  wwlksn  29356  submomnd  32496  suborng  32701  bj-restsnid  36273  1aryenef  47420  2aryenef  47431
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