Theorem f1ofn 5147

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

Assertion

Ref	Expression
f1ofn	⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)

Proof of Theorem f1ofn

Step	Hyp	Ref	Expression
1		f1of 5146	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5066	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   Fn wfn 4917  ⟶wf 4918  –1-1-onto→wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-f 4926  df-f1 4927  df-f1o 4929

This theorem is referenced by:  f1ofun  5148  f1odm  5150  isocnv2  5472  isoini  5477  isoselem  5479  bren  6251  en1  6302  phplem4  6341  phplem4on  6353  dif1en  6364  supisolem  6421  ordiso2  6446