Theorem f1fun 6016

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 6015	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fnfun 5902	. 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 5798   Fn wfn 5799  –1-1→wf1 5801

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 196  df-an 385  df-fn 5807  df-f 5808  df-f1 5809

This theorem is referenced by:  f1cocnv2  6077  f1o2ndf1  7172  fnwelem  7179  f1dmvrnfibi  8133  fsuppco  8190  ackbij1b  8944  fin23lem31  9048  fin1a2lem6  9110  hashimarn  13085  gsumval3lem1  18129  gsumval3lem2  18130  usgrafun  25878  elhf  31451  usgrfun  40388  trlsegvdeglem6  41393