Theorem fvfundmfvn0 6883

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	⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Proof of Theorem fvfundmfvn0

Step	Hyp	Ref	Expression
1		ndmfv 6875	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
2	1	necon1ai 2952	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3		nfunsn 6882	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
4	3	necon1ai 2952	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴}))
5	2, 4	jca 511	1 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ≠ wne 2925  ∅c0 4292  {csn 4585  dom cdm 5631   ↾ cres 5633  Fun wfun 6493  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6452  df-fun 6501  df-fv 6507

This theorem is referenced by:  fvn0ssdmfun  7028  feldmfvelcdm  7040  fvn0fvelrnOLD  7117  umgrnloopv  29086  usgrnloopvALT  29181  afvpcfv0  47140  afvfvn0fveq  47144  afv0nbfvbi  47145  afv2fvn0fveq  47258  ovn0dmfun  48137  reldmlan2  49599  reldmran2  49600  lanrcl  49603  ranrcl  49604  lmdran  49653  cmdlan  49654