Theorem fvfundmfvn0 6963

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 6955	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
2	1	necon1ai 2974	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3		nfunsn 6962	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
4	3	necon1ai 2974	. 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 2108   ≠ wne 2946  ∅c0 4352  {csn 4648  dom cdm 5700   ↾ cres 5702  Fun wfun 6567  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581

This theorem is referenced by:  fvn0ssdmfun  7108  feldmfvelcdm  7120  fvn0fvelrnOLD  7197  umgrnloopv  29141  usgrnloopvALT  29236  afvpcfv0  47061  afvfvn0fveq  47065  afv0nbfvbi  47066  afv2fvn0fveq  47179  ovn0dmfun  47879