Theorem f1fn 5405

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 5403	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		ffn 5347	. 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 5193  ⟶wf 5194  –1-1→wf1 5195

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 5202  df-f1 5203

This theorem is referenced by:  f1fun  5406  f1rel  5407  f1dm  5408  f1ssr  5410  f1f1orn  5453  f1elima  5752  f1eqcocnv  5770  f1oiso  5805  phplem4dom  6840  f1finf1o  6924  updjudhcoinlf  7057  updjudhcoinrg  7058  updjud  7059  fihashf1rn  10723