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Theorem fvn0fvelrn 6938
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 6937 . 2 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 nelsn 4671 . 2 ((𝐹𝑋) ≠ ∅ → ¬ (𝐹𝑋) ∈ {∅})
3 elunnel2 4165 . 2 (((𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹𝑋) ∈ {∅}) → (𝐹𝑋) ∈ ran 𝐹)
41, 2, 3sylancr 587 1 ((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  wne 2938  cun 3961  c0 4339  {csn 4631  ran crn 5690  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571
This theorem is referenced by:  elfvunirn  6939  wlkvtxiedg  29658
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