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Theorem ovprc 6723
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 6693 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 df-br 4686 . . . . 5 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3 ovprc1.1 . . . . . 6 Rel dom 𝐹
4 brrelex12 5189 . . . . . 6 ((Rel dom 𝐹𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
53, 4mpan 706 . . . . 5 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
62, 5sylbir 225 . . . 4 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
76con3i 150 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8 ndmfv 6256 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
97, 8syl 17 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
101, 9syl5eq 2697 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  cop 4216   class class class wbr 4685  dom cdm 5143  Rel wrel 5148  cfv 5926  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  ovprc1  6724  ovprc2  6725  ovrcl  6726  elbasov  15968  firest  16140  psrplusg  19429  psrmulr  19432  psrvscafval  19438  mplval  19476  opsrle  19523  opsrbaslem  19525  opsrbaslemOLD  19526  evlval  19572  matbas0pc  20263  mdetfval  20440  madufval  20491  mdegfval  23867  nbgrprc0  26271  brovmptimex  38642
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