Theorem ovprc 7394

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

Step	Hyp	Ref	Expression
1		df-ov 7359	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		df-br 5073	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3		ovprc1.1	. . . . 5 ⊢ Rel dom 𝐹
4	3	brrelex12i 5673	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	2, 4	sylbir 236	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		ndmfv 6859	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
7	5, 6	nsyl5 159	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
8	1, 7	eqtrid 2786	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4261  ⟨cop 4561   class class class wbr 5072  dom cdm 5618  Rel wrel 5623  ‘cfv 6485  (class class class)co 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-dm 5628  df-iota 6441  df-fv 6493  df-ov 7359

This theorem is referenced by:  ovprc1  7395  ovprc2  7396  ovrcl  7397  elbasov  17177  firest  17386  psrplusg  21912  psrmulr  21917  psrvscafval  21923  mplval  21963  opsrle  22023  opsrbaslem  22025  evlval  22076  matbas0pc  22392  mdetfval  22569  madufval  22620  mdegfval  26045  nbgrprc0  29421  gonan0  35620  brovmptimex  44471  clnbgrprc0  48311  gricrcl  48405  grlicrcl  48498  grilcbri2  48502  upfval  49666  reldmprcof1  49871  reldmprcof2  49872  lmdfval  50139  cmdfval  50140