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Theorem fvn0fvelrn 6923
Description: If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by SN, 13-Jan-2025.)
Assertion
Ref Expression
fvn0fvelrn ((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)

Proof of Theorem fvn0fvelrn
StepHypRef Expression
1 fvrn0 6922 . 2 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 nelsn 4664 . 2 ((𝐹𝑋) ≠ ∅ → ¬ (𝐹𝑋) ∈ {∅})
3 elunnel2 4143 . 2 (((𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹𝑋) ∈ {∅}) → (𝐹𝑋) ∈ ran 𝐹)
41, 2, 3sylancr 585 1 ((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2098  wne 2930  cun 3937  c0 4318  {csn 4624  ran crn 5673  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683  df-iota 6495  df-fv 6551
This theorem is referenced by:  elfvunirn  6924  wlkvtxiedg  29483
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