Theorem ffrn 6676

Description: A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

Ref	Expression
ffrn	⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)

Proof of Theorem ffrn

Step	Hyp	Ref	Expression
1		ffn 6663	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn3 6675	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 218	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   → wi 4  ran crn 5626   Fn wfn 6488  ⟶wf 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797

This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3907  df-f 6497

This theorem is referenced by:  fo2ndf  8065  mapsnd  8828  itg1val2  25664  esplyfval1  33735  aks6d1c6lem3  42628  volicoff  46444  f1cof1b  47540  f1ocof1ob  47544  fundcmpsurbijinjpreimafv  47882