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Theorem fvn0fvelrn 6871
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 6870 . 2 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 nelsn 4625 . 2 ((𝐹𝑋) ≠ ∅ → ¬ (𝐹𝑋) ∈ {∅})
3 elunnel2 4109 . 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 2942  cun 3907  c0 4281  {csn 4585  ran crn 5633  cfv 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-cnv 5640  df-dm 5642  df-rn 5643  df-iota 6446  df-fv 6502
This theorem is referenced by:  elfvunirn  6872  wlkvtxiedg  28471
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