Theorem f1fn 5330

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 5328	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5272	. 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 5118  ⟶wf 5119  –1-1→wf1 5120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-f 5127  df-f1 5128

This theorem is referenced by:  f1fun  5331  f1rel  5332  f1dm  5333  f1ssr  5335  f1f1orn  5378  f1elima  5674  f1eqcocnv  5692  f1oiso  5727  phplem4dom  6756  f1finf1o  6835  updjudhcoinlf  6965  updjudhcoinrg  6966  updjud  6967  fihashf1rn  10535