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Theorem fvfundmfvn0 6701
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 6693 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
21necon1ai 3040 . 2 ((𝐹𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3 nfunsn 6700 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
43necon1ai 3040 . 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 2105  wne 3013  c0 4288  {csn 4557  dom cdm 5548  cres 5550  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  fvn0ssdmfun  6834  fvn0fvelrn  6917  umgrnloopv  26818  usgrnloopvALT  26910  afvpcfv0  43222  afvfvn0fveq  43226  afv0nbfvbi  43227  afv2fvn0fveq  43340  ovn0dmfun  43908
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