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Theorem funeldmb 33033
Description: If is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.)
Assertion
Ref Expression
funeldmb ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ≠ ∅))

Proof of Theorem funeldmb
StepHypRef Expression
1 fvelrn 6833 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
21ex 416 . . . . . . 7 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
32adantr 484 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
4 eleq1 2903 . . . . . . 7 ((𝐹𝐴) = ∅ → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
54adantl 485 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
63, 5sylibd 242 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹))
76con3d 155 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹))
87impancom 455 . . 3 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹))
9 ndmfv 6689 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
108, 9impbid1 228 . 2 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
1110necon2abid 3056 1 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  c0 4276  dom cdm 5543  ran crn 5544  Fun wfun 6338  cfv 6344
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352
This theorem is referenced by:  nosepssdm  33217
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