Theorem fvn0fvelrn 6912

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 6911	. 2 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		nelsn 4633	. 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → ¬ (𝐹‘𝑋) ∈ {∅})
3		elunnel2 4110	. 2 ⊢ (((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹‘𝑋) ∈ {∅}) → (𝐹‘𝑋) ∈ ran 𝐹)
4	1, 2, 3	sylancr 598	1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2143   ≠ wne 2958   ∪ cun 3904  ∅c0 4287  {csn 4590  ran crn 5664  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-cnv 5671  df-dm 5673  df-rn 5674  df-iota 6494  df-fv 6546

This theorem is referenced by:  elfvunirn  6913  wlkvtxiedg  29955