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Theorem fvfundmfvn0 6934
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 6926 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
21necon1ai 2958 . 2 ((𝐹𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3 nfunsn 6933 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
43necon1ai 2958 . 2 ((𝐹𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴}))
52, 4jca 510 1 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wne 2930  c0 4318  {csn 4624  dom cdm 5672  cres 5674  Fun wfun 6536  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550
This theorem is referenced by:  fvn0ssdmfun  7078  feldmfvelcdm  7090  fvn0fvelrnOLD  7167  umgrnloopv  28961  usgrnloopvALT  29056  afvpcfv0  46588  afvfvn0fveq  46592  afv0nbfvbi  46593  afv2fvn0fveq  46706  ovn0dmfun  47329
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