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Theorem ovprc2 6917
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 478 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
21con3i 152 . 2 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 ovprc1.1 . . 3 Rel dom 𝐹
43ovprc 6915 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
52, 4syl 17 1 𝐵 ∈ V → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  c0 4115  dom cdm 5312  Rel wrel 5317  (class class class)co 6878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-dm 5322  df-iota 6064  df-fv 6109  df-ov 6881
This theorem is referenced by:  ressbasss  16257  ress0  16259  wunress  16266  0rest  16405  firest  16408  subcmn  18557  dprdval0prc  18717  psrbas  19701  psr1val  19878  vr1val  19884  ply1ascl  19950  evl1fval  20014  zrhval  20178  dsmmval2  20405  restbas  21291  resstopn  21319  deg1fval  24181  wwlksn  27088  clwwlknOLD  27331  submomnd  30226  suborng  30331  bj-restsnid  33533
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