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Theorem funeldmb 31787
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 6392 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
21ex 449 . . . . . . 7 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
32adantr 480 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ ran 𝐹))
4 eleq1 2718 . . . . . . 7 ((𝐹𝐴) = ∅ → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
54adantl 481 . . . . . 6 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → ((𝐹𝐴) ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
63, 5sylibd 229 . . . . 5 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐴 ∈ dom 𝐹 → ∅ ∈ ran 𝐹))
76con3d 148 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (¬ ∅ ∈ ran 𝐹 → ¬ 𝐴 ∈ dom 𝐹))
87impancom 455 . . 3 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ → ¬ 𝐴 ∈ dom 𝐹))
9 ndmfv 6256 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
108, 9impbid1 215 . 2 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → ((𝐹𝐴) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐹))
1110necon2abid 2865 1 ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  c0 3948  dom cdm 5143  ran crn 5144  Fun wfun 5920  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  nosepssdm  31961
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