Theorem f1ofun 5303

Description: A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Assertion

Ref	Expression
f1ofun	⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)

Proof of Theorem f1ofun

Step	Hyp	Ref	Expression
1		f1ofn 5302	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fnfun 5156	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)

Syntax hints:   → wi 4  Fun wfun 5053   Fn wfn 5054  –1-1-onto→wf1o 5058

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-fn 5062  df-f 5063  df-f1 5064  df-f1o 5066

This theorem is referenced by:  f1orel  5304  f1oresrab  5517  isose  5654  f1opw  5909  xpcomco  6649  fiintim  6746  f1dmvrnfibi  6760  caseinl  6891  caseinr  6892  ctssdclemr  6911  fihasheqf1oi  10375  fisumss  11000  ennnfonelemex  11719  ennnfonelemf1  11723