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Theorem aovvdm 45729
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 45665 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
21eleq1i 2824 . 2 ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3 afvvdm 45685 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
42, 3sylbi 216 1 ( ((𝐴𝐹𝐵)) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cop 4629  dom cdm 5670  '''cafv 45661   ((caov 45662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4945  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-res 5682  df-iota 6485  df-fun 6535  df-fv 6541  df-aiota 45629  df-dfat 45663  df-afv 45664  df-aov 45665
This theorem is referenced by:  ndmaovrcl  45748  ndmaovass  45750  ndmaovdistr  45751
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