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Theorem afvvdm 44087
 Description: If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvvdm ((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)

Proof of Theorem afvvdm
StepHypRef Expression
1 ndmafv 44086 . . 3 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)
2 nvelim 44069 . . 3 ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵)
31, 2syl 17 . 2 𝐴 ∈ dom 𝐹 → ¬ (𝐹'''𝐴) ∈ 𝐵)
43con4i 114 1 ((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  dom cdm 5524  '''cafv 44063 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-res 5536  df-iota 6294  df-fun 6337  df-fv 6343  df-aiota 44030  df-dfat 44065  df-afv 44066 This theorem is referenced by:  aovvdm  44131  aovrcl  44135  aoprssdm  44148
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