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Theorem aovvdm 41586
 Description: If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovvdm ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)

Proof of Theorem aovvdm
StepHypRef Expression
1 df-aov 41519 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
21eleq1i 2721 . 2 ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3 afvvdm 41542 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
42, 3sylbi 207 1 ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030  ⟨cop 4216  dom cdm 5143  '''cafv 41515   ((caov 41516 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 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-un 3612  df-if 4120  df-fv 5934  df-dfat 41517  df-afv 41518  df-aov 41519 This theorem is referenced by:  ndmaovrcl  41605  ndmaovass  41607  ndmaovdistr  41608
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