Theorem funeldmb 7307

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 7022	. . . . . . . 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 2825	. . . . . . 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 6866	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
10	8, 9	impbid1 225	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
11	10	necon2abid 2975	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 2933  ∅c0 4274  dom cdm 5624  ran crn 5625  Fun wfun 6486  ‘cfv 6492

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 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  nosepssdm  27664