Theorem funeldmb 7314

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 7028	. . . . . . . 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 2824	. . . . . . 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 6872	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
10	8, 9	impbid1 225	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
11	10	necon2abid 2974	1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∅c0 4273  dom cdm 5631  ran crn 5632  Fun wfun 6492  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506

This theorem is referenced by:  nosepssdm  27650