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Theorem funeldmb 32204
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 6602 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
21ex 403 . . . . . . 7 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
32adantr 474 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
4 eleq1 2895 . . . . . . 7 ((𝐹𝐴) = ∅ → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
54adantl 475 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
63, 5sylibd 231 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹))
76con3d 150 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹))
87impancom 445 . . 3 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹))
9 ndmfv 6464 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
108, 9impbid1 217 . 2 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
1110necon2abid 3042 1 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wne 3000  c0 4145  dom cdm 5343  ran crn 5344  Fun wfun 6118  cfv 6124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fn 6127  df-fv 6132
This theorem is referenced by:  nosepssdm  32376
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