Theorem ffrn 6723

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 6709	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn3 6722	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 221	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)

This proof depends on syntax axioms:   → wi 4  ran crn 5652   Fn wfn 6533  ⟶wf 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828

This proof depends on definitions:  df-bi 210  df-an 402  df-ss 3916  df-f 6542

This theorem is used by:  fo2ndf  8132  mapsnd  8914  itg1val2  26005  esplyfval1  34205  aks6d1c6lem3  43222  volicoff  47004  f1cof1b  48146  f1ocof1ob  48150  fundcmpsurbijinjpreimafv  48488