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Theorem fvn0fvelrn 6874
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 6873 . 2 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 nelsn 4627 . 2 ((𝐹𝑋) ≠ ∅ → ¬ (𝐹𝑋) ∈ {∅})
3 elunnel2 4111 . 2 (((𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹𝑋) ∈ {∅}) → (𝐹𝑋) ∈ ran 𝐹)
41, 2, 3sylancr 588 1 ((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107  wne 2940  cun 3909  c0 4283  {csn 4587  ran crn 5635  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-iota 6449  df-fv 6505
This theorem is referenced by:  elfvunirn  6875  wlkvtxiedg  28615
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