Theorem fnrndomg

Description: The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004)

	Ref	Expression
Assertion	fnrndomg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dffn4	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 )
2		fodomg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐹 : 𝐴 –onto→ ran 𝐹 → ran 𝐹 ≼ 𝐴 ) )
3	1 2	syl5bi	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴 ) )