Theorem ovn0dmfun 45318

Description: If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6812. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
ovn0dmfun	⊢ ((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))

Proof of Theorem ovn0dmfun

Step	Hyp	Ref	Expression
1		df-ov 7278	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	neeq1i 3008	. 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅)
3		fvfundmfvn0 6812	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))
4	2, 3	sylbi 216	1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  {csn 4561  ⟨cop 4567  dom cdm 5589   ↾ cres 5591  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278

This theorem is referenced by: (None)