Theorem fof

Description: An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994)

	Ref	Expression
Assertion	fof	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqimss	⊢ ( ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵 )
2	1	anim2i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
3		df-fo	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) )
4		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
5	2 3 4	3imtr4i	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )