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Theorem ovprc2 6650
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 477 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
21con3i 150 . 2 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 ovprc1.1 . . 3 Rel dom 𝐹
43ovprc 6648 . 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 384   = wceq 1480  wcel 1987  Vcvv 3190  c0 3897  dom cdm 5084  Rel wrel 5089  (class class class)co 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-dm 5094  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  ressbasss  15872  ress0  15874  wunress  15880  0rest  16030  firest  16033  subcmn  18182  dprdval0prc  18341  psrbas  19318  psr1val  19496  vr1val  19502  ply1ascl  19568  evl1fval  19632  zrhval  19796  dsmmval2  20020  restbas  20902  resstopn  20930  deg1fval  23778  wwlksn  26632  wwlks2onv  26750  clwwlksn  26782  submomnd  29537  suborng  29642  bj-restsnid  32730
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