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Theorem ovprc2 7180
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 488 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2 ovprc1.1 . . 3 Rel dom 𝐹
32ovprc 7178 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
41, 3nsyl5 162 1 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265  dom cdm 5532  Rel wrel 5537  (class class class)co 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-dm 5542  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  ressbasss  16547  ress0  16549  wunress  16555  0rest  16694  firest  16697  subcmn  18948  dprdval0prc  19115  zrhval  20199  dsmmval2  20423  psrbas  20614  psr1val  20813  vr1val  20819  ply1ascl  20885  evl1fval  20950  restbas  21761  resstopn  21789  deg1fval  24679  wwlksn  27621  submomnd  30742  suborng  30920  bj-restsnid  34463  1aryenef  44999  2aryenef  45010
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