Theorem fvn0fvelrn 6860

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	⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

Proof of Theorem fvn0fvelrn

Step	Hyp	Ref	Expression
1		fvrn0 6859	. 2 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		nelsn 4620	. 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → ¬ (𝐹‘𝑋) ∈ {∅})
3		elunnel2 4106	. 2 ⊢ (((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹‘𝑋) ∈ {∅}) → (𝐹‘𝑋) ∈ ran 𝐹)
4	1, 2, 3	sylancr 587	1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113   ≠ wne 2930   ∪ cun 3897  ∅c0 4284  {csn 4577  ran crn 5622  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-cnv 5629  df-dm 5631  df-rn 5632  df-iota 6445  df-fv 6497

This theorem is referenced by:  elfvunirn  6861  wlkvtxiedg  29614