Theorem f1fun 5604

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 5603	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fnfun 5481	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)

This proof depends on syntax axioms:   → wi 4  Fun wfun 5374   Fn wfn 5375  –1-1→wf1 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117  df-fn 5383  df-f 5384  df-f1 5385

This theorem is used by:  f1cocnv2  5670  f1o2ndf1  6468  f1dmvrnfibi  7262  fsuppcorn  7306  djuinj  7452  usgrfun  16607