Theorem f1fun 6103

Description: A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
f1fun	⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)

Proof of Theorem f1fun

Step	Hyp	Ref	Expression
1		f1fn 6102	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fnfun 5988	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	1, 2	syl 17	1 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)

Syntax hints:   → wi 4  Fun wfun 5882   Fn wfn 5883  –1-1→wf1 5885

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-fn 5891  df-f 5892  df-f1 5893

This theorem is referenced by:  f1cocnv2  6164  f1o2ndf1  7285  fnwelem  7292  f1dmvrnfibi  8250  fsuppco  8307  ackbij1b  9061  fin23lem31  9165  fin1a2lem6  9227  hashimarn  13227  gsumval3lem1  18306  gsumval3lem2  18307  usgrfun  26053  trlsegvdeglem6  27085  elhf  32281