Theorem fodomg

Description: An onto function implies dominance of domain over range. Lemma 10.20 of Kunen p. 30. This theorem uses the axiom of choice ac7g . The axiom of choice is not needed for finite sets, see fodomfi . See also fodomnum . (Contributed by NM, 23-Jul-2004) (Proof shortened by BJ, 20-May-2024)

	Ref	Expression
Assertion	fodomg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ≼ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		numth3	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ dom card )
2		fodomnum	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ≼ 𝐴 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ≼ 𝐴 ) )