MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funeldmb Structured version   Visualization version   GIF version

Theorem funeldmb 7379
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 7096 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
21ex 412 . . . . . . 7 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
32adantr 480 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
4 eleq1 2829 . . . . . . 7 ((𝐹𝐴) = ∅ → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
54adantl 481 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
63, 5sylibd 239 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹))
76con3d 152 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹))
87impancom 451 . . 3 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹))
9 ndmfv 6941 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
108, 9impbid1 225 . 2 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
1110necon2abid 2983 1 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333  dom cdm 5685  ran crn 5686  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  nosepssdm  27731
  Copyright terms: Public domain W3C validator