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Theorem ovprc2 7311
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 485 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2 ovprc1.1 . . 3 Rel dom 𝐹
32ovprc 7309 . 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 396   = wceq 1542  wcel 2110  Vcvv 3431  c0 4262  dom cdm 5590  Rel wrel 5595  (class class class)co 7271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-dm 5600  df-iota 6390  df-fv 6440  df-ov 7274
This theorem is referenced by:  ressbasss  16948  ress0  16951  wunress  16958  wunressOLD  16959  0rest  17138  firest  17141  subcmn  19436  dprdval0prc  19603  zrhval  20707  dsmmval2  20941  psrbas  21145  psr1val  21355  vr1val  21361  ply1ascl  21427  evl1fval  21492  restbas  22307  resstopn  22335  deg1fval  25243  wwlksn  28198  submomnd  31332  suborng  31510  bj-restsnid  35254  1aryenef  45960  2aryenef  45971
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