Theorem f1ofun 5463

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 5462	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fnfun 5313	. 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 5210   Fn wfn 5211  –1-1-onto→wf1o 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-fn 5219  df-f 5220  df-f1 5221  df-f1o 5223

This theorem is referenced by:  f1orel  5464  f1oresrab  5681  isose  5821  f1opw  6077  xpcomco  6825  fiintim  6927  f1dmvrnfibi  6942  caseinl  7089  caseinr  7090  ctssdccl  7109  ctssdclemr  7110  fihasheqf1oi  10766  fisumss  11399  ennnfonelemex  12414  ennnfonelemf1  12418  hmeontr  13783