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Theorem ovprc 6827
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 6795 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 df-br 4785 . . . . 5 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3 ovprc1.1 . . . . . 6 Rel dom 𝐹
4 brrelex12 5295 . . . . . 6 ((Rel dom 𝐹𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
53, 4mpan 662 . . . . 5 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
62, 5sylbir 225 . . . 4 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
76con3i 151 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8 ndmfv 6359 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
97, 8syl 17 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
101, 9syl5eq 2816 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  cop 4320   class class class wbr 4784  dom cdm 5249  Rel wrel 5254  cfv 6031  (class class class)co 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259  df-iota 5994  df-fv 6039  df-ov 6795
This theorem is referenced by:  ovprc1  6828  ovprc2  6829  ovrcl  6830  elbasov  16127  firest  16300  psrplusg  19595  psrmulr  19598  psrvscafval  19604  mplval  19642  opsrle  19689  opsrbaslem  19691  opsrbaslemOLD  19692  evlval  19738  matbas0pc  20431  mdetfval  20609  madufval  20660  mdegfval  24041  nbgrprc0  26448  brovmptimex  38844
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