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Theorem fvfundmfvn0 6890
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 6882 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
21necon1ai 2972 . 2 ((𝐹𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3 nfunsn 6889 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
43necon1ai 2972 . 2 ((𝐹𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴}))
52, 4jca 513 1 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2944  c0 4287  {csn 4591  dom cdm 5638  cres 5640  Fun wfun 6495  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  fvn0ssdmfun  7030  fvn0fvelrnOLD  7114  umgrnloopv  28099  usgrnloopvALT  28191  afvpcfv0  45452  afvfvn0fveq  45456  afv0nbfvbi  45457  afv2fvn0fveq  45570  ovn0dmfun  46132
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