Theorem fvn0fvelrn 6890

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 6889	. 2 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		nelsn 4624	. 2 ⊢ ((𝐹‘𝑋) ≠ ∅ → ¬ (𝐹‘𝑋) ∈ {∅})
3		elunnel2 4108	. 2 ⊢ (((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) ∧ ¬ (𝐹‘𝑋) ∈ {∅}) → (𝐹‘𝑋) ∈ ran 𝐹)
4	1, 2, 3	sylancr 596	1 ⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2141   ≠ wne 2956   ∪ cun 3902  ∅c0 4285  {csn 4581  ran crn 5646  ‘cfv 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-cnv 5653  df-dm 5655  df-rn 5656  df-iota 6471  df-fv 6523

This theorem is referenced by:  elfvunirn  6891  wlkvtxiedg  29769