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Theorem ovprc 7189
Description: The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
ovprc1.1 Rel dom 𝐹
Assertion
Ref Expression
ovprc (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem ovprc
StepHypRef Expression
1 df-ov 7154 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 df-br 5054 . . . 4 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3 ovprc1.1 . . . . 5 Rel dom 𝐹
43brrelex12i 5595 . . . 4 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
52, 4sylbir 238 . . 3 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6 ndmfv 6693 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
75, 6nsyl5 162 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
81, 7syl5eq 2871 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276  cop 4556   class class class wbr 5053  dom cdm 5543  Rel wrel 5548  cfv 6345  (class class class)co 7151
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-xp 5549  df-rel 5550  df-dm 5553  df-iota 6304  df-fv 6353  df-ov 7154
This theorem is referenced by:  ovprc1  7190  ovprc2  7191  ovrcl  7192  elbasov  16547  firest  16708  psrplusg  20628  psrmulr  20631  psrvscafval  20637  mplval  20675  opsrle  20724  opsrbaslem  20726  evlval  20776  matbas0pc  21023  mdetfval  21200  madufval  21251  mdegfval  24672  nbgrprc0  27133  gonan0  32724  brovmptimex  40677
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