Theorem ffrn 6516

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 6503	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		dffn3 6515	. 2 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
3	1, 2	sylib 221	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   → wi 4  ran crn 5529   Fn wfn 6335  ⟶wf 6336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877  df-f 6344

This theorem is referenced by:  fo2ndf  7828  mapsnd  8481  itg1val2  24398  selvval2lem4  39774  volicoff  43048  fundcmpsurbijinjpreimafv  44351