Theorem ovprc 7447

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 7412	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		df-br 5150	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3		ovprc1.1	. . . . 5 ⊢ Rel dom 𝐹
4	3	brrelex12i 5732	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	2, 4	sylbir 234	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		ndmfv 6927	. . 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 2785	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  ⟨cop 4635   class class class wbr 5149  dom cdm 5677  Rel wrel 5682  ‘cfv 6544  (class class class)co 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412

This theorem is referenced by:  ovprc1  7448  ovprc2  7449  ovrcl  7450  elbasov  17151  firest  17378  psrplusg  21500  psrmulr  21503  psrvscafval  21509  mplval  21548  opsrle  21602  opsrbaslem  21604  opsrbaslemOLD  21605  evlval  21658  matbas0pc  21909  mdetfval  22088  madufval  22139  mdegfval  25580  nbgrprc0  28591  gonan0  34383  brovmptimex  42778