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Theorem fvfundmfvn0 6882
Description: If the "value of a class" at an argument is not the empty set, then the argument is in the domain of the class and the class restricted to the singleton formed on that argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) (Proof shortened by BJ, 13-Aug-2022.)
Assertion
Ref Expression
fvfundmfvn0 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Proof of Theorem fvfundmfvn0
StepHypRef Expression
1 ndmfv 6874 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
21necon1ai 2969 . 2 ((𝐹𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3 nfunsn 6881 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
43necon1ai 2969 . 2 ((𝐹𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴}))
52, 4jca 512 1 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wne 2941  c0 4280  {csn 4584  dom cdm 5631  cres 5633  Fun wfun 6487  cfv 6493
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6445  df-fun 6495  df-fv 6501
This theorem is referenced by:  fvn0ssdmfun  7022  fvn0fvelrnOLD  7105  umgrnloopv  27943  usgrnloopvALT  28035  afvpcfv0  45310  afvfvn0fveq  45314  afv0nbfvbi  45315  afv2fvn0fveq  45428  ovn0dmfun  45990
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