Theorem fnrndomnum

Description: A version of fnrndomg that does not require the axiom of choice ax-ac . (Contributed by Vincent Gonzalez, 17-Aug-2026)

	Ref	Expression
Assertion	fnrndomnum	⊢ ( 𝐴 ∈ dom card → ( 𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dffn4	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 )
2		fodomnum	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ ran 𝐹 → ran 𝐹 ≼ 𝐴 ) )
3	1 2	biimtrid	⊢ ( 𝐴 ∈ dom card → ( 𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴 ) )