Theorem fvfundmfvn0 6871

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 6863	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
2	1	necon1ai 2963	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3		nfunsn 6870	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
4	3	necon1ai 2963	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴}))
5	2, 4	jca 517	1 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2121   ≠ wne 2936  ∅c0 4264  {csn 4558  dom cdm 5621   ↾ cres 5623  Fun wfun 6483  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-iota 6445  df-fun 6491  df-fv 6497

This theorem is referenced by:  fvn0ssdmfun  7019  feldmfvelcdm  7031  umgrnloopv  29197  usgrnloopvALT  29292  afvpcfv0  47623  afvfvn0fveq  47627  afv0nbfvbi  47628  afv2fvn0fveq  47741  ovn0dmfun  48661  reldmlan2  50121  reldmran2  50122  lanrcl  50125  ranrcl  50126  lmdran  50175  cmdlan  50176