Theorem ovn0dmfun 48648

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 6874. (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 7366	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	neeq1i 2999	. 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅)
3		fvfundmfvn0 6874	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))
4	2, 3	sylbi 218	1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119   ≠ wne 2935  ∅c0 4268  {csn 4562  ⟨cop 4568  dom cdm 5625   ↾ cres 5627  Fun wfun 6486  ‘cfv 6492  (class class class)co 7363

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 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

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 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366

This theorem is referenced by: (None)