Theorem f1fn 5266

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 5264	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5208	. 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 5054  ⟶wf 5055  –1-1→wf1 5056

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-f 5063  df-f1 5064

This theorem is referenced by:  f1fun  5267  f1rel  5268  f1dm  5269  f1ssr  5271  f1f1orn  5312  f1elima  5606  f1eqcocnv  5624  f1oiso  5659  phplem4dom  6685  f1finf1o  6763  updjudhcoinlf  6880  updjudhcoinrg  6881  updjud  6882  fihashf1rn  10376