Theorem fnfvelrnd 42808

Description: A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
fnfvelrnd.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnfvelrnd.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
fnfvelrnd	⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)

Proof of Theorem fnfvelrnd

Step	Hyp	Ref	Expression
1		fnfvelrnd.1	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fnfvelrnd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		fnfvelrn 6958	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∈ wcel 2106  ran crn 5590   Fn wfn 6428  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  limsupgtlem  43318