Theorem fnrndomg

Description: The range of a function is dominated by its domain. This theorem requires the axiom of choice ax-ac2 ; see fnrndomnum for a version that does not. (Contributed by NM, 1-Sep-2004) (Proof shortened by Vincent Gonzalez, 17-Aug-2026)

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

Proof

Step	Hyp	Ref	Expression
1		numth3	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ dom card )
2		fnrndomnum	⊢ ( 𝐴 ∈ dom card → ( 𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴 ) )