Theorem funeldmb 7379

Description: If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.)

Assertion

Ref	Expression
funeldmb	⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))

Proof of Theorem funeldmb

Step	Hyp	Ref	Expression
1		fvelrn 7096	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)
2	1	ex 412	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ ran 𝐹))
3	2	adantr 480	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ ran 𝐹))
4		eleq1 2827	. . . . . . 7 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
5	4	adantl 481	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → ((𝐹‘𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
6	3, 5	sylibd 239	. . . . 5 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹))
7	6	con3d 152	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹))
8	7	impancom 451	. . 3 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹))
9		ndmfv 6942	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
10	8, 9	impbid1 225	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
11	10	necon2abid 2981	1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∅c0 4339  dom cdm 5689  ran crn 5690  Fun wfun 6557  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571

This theorem is referenced by:  nosepssdm  27746