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Theorem ovprc2 7459
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 7457 . 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 1533  wcel 2098  Vcvv 3461  c0 4322  dom cdm 5678  Rel wrel 5683  (class class class)co 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-dm 5688  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  ressbasssg  17220  ressbasssOLD  17223  ress0  17227  wunress  17234  wunressOLD  17235  0rest  17414  firest  17417  subcmn  19804  dprdval0prc  19971  zrhval  21450  dsmmval2  21687  psrbas  21895  psr1val  22128  vr1val  22134  ply1ascl  22202  evl1fval  22272  restbas  23106  resstopn  23134  deg1fval  26060  wwlksn  29720  submomnd  32880  suborng  33129  bj-restsnid  36694  1aryenef  47901  2aryenef  47912
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