Theorem f1fn 5461

Description: A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
f1fn	⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)

Proof of Theorem f1fn

Step	Hyp	Ref	Expression
1		f1f 5459	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5403	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3	1, 2	syl 14	1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   Fn wfn 5249  ⟶wf 5250  –1-1→wf1 5251

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-f 5258  df-f1 5259

This theorem is referenced by:  f1fun  5462  f1rel  5463  f1dm  5464  f1ssr  5466  f1f1orn  5511  f1elima  5816  f1eqcocnv  5834  f1oiso  5869  phplem4dom  6918  f1finf1o  7006  updjudhcoinlf  7139  updjudhcoinrg  7140  updjud  7141  fihashf1rn  10859  kerf1ghm  13344