Theorem ffrn 6598

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 6584	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn3 6597	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 217	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   → wi 4  ran crn 5581   Fn wfn 6413  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-f 6422

This theorem is referenced by:  fo2ndf  7933  mapsnd  8632  itg1val2  24753  selvval2lem4  40154  volicoff  43426  f1cof1b  44456  f1ocof1ob  44460  fundcmpsurbijinjpreimafv  44747