Theorem fvfundmfvn0 6945

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 6937	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
2	1	necon1ai 2986	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹)
3		nfunsn 6944	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
4	3	necon1ai 2986	. 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴}))
5	2, 4	jca 521	1 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  {csn 4588  dom cdm 5655   ↾ cres 5657  Fun wfun 6549  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6509  df-fun 6557  df-fv 6563

This theorem is used by:  fvn0ssdmfun  7095  feldmfvelcdm  7107  umgrnloopv  29711  usgrnloopvALT  29809  kardcard2a  35931  afvpcfv0  48419  afvfvn0fveq  48423  afv0nbfvbi  48424  afv2fvn0fveq  48537  ovn0dmfun  49457  reldmlan2  51003  reldmran2  51004  lanrcl  51007  ranrcl  51008  lmdran  51057  cmdlan  51058