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Theorem ovprc2 7449
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 486 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2 ovprc1.1 . . 3 Rel dom 𝐹
32ovprc 7447 . 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 397   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323  dom cdm 5677  Rel wrel 5682  (class class class)co 7409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 7412
This theorem is referenced by:  ressbasssg  17181  ressbasssOLD  17184  ress0  17188  wunress  17195  wunressOLD  17196  0rest  17375  firest  17378  subcmn  19705  dprdval0prc  19872  zrhval  21057  dsmmval2  21291  psrbas  21497  psr1val  21710  vr1val  21716  ply1ascl  21780  evl1fval  21847  restbas  22662  resstopn  22690  deg1fval  25598  wwlksn  29091  submomnd  32228  suborng  32433  bj-restsnid  35968  1aryenef  47331  2aryenef  47342
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