Theorem funeldmb 7400

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 7108	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)
2	1	ex 418	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ ran 𝐹))
3	2	adantr 486	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ ran 𝐹))
4		eleq1 2852	. . . . . . 7 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
5	4	adantl 487	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → ((𝐹‘𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
6	3, 5	sylibd 242	. . . . 5 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹))
7	6	con3d 153	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝐹‘𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹))
8	7	impancom 457	. . 3 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹))
9		ndmfv 6948	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
10	8, 9	impbid1 228	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
11	10	necon2abid 3001	1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  dom cdm 5658  ran crn 5659  Fun wfun 6560  ‘cfv 6566

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-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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-nf 1817  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 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574

This theorem is used by:  nosepssdm  28100