Theorem f1ofun 5506

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 5505	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fnfun 5355	. 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 5252   Fn wfn 5253  –1-1-onto→wf1o 5257

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 5261  df-f 5262  df-f1 5263  df-f1o 5265

This theorem is referenced by:  f1orel  5507  f1oresrab  5727  isose  5868  f1opw  6130  xpcomco  6885  fiintim  6992  f1dmvrnfibi  7010  caseinl  7157  caseinr  7158  ctssdccl  7177  ctssdclemr  7178  fihasheqf1oi  10879  fisumss  11557  ennnfonelemex  12631  ennnfonelemf1  12635  hmeontr  14549