Theorem brdomi

Description: Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015) Avoid ax-un . (Revised by BTernaryTau, 29-Nov-2024)

	Ref	Expression
Assertion	brdomi	⊢ ( 𝐴 ≼ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		reldom	⊢ Rel ≼
2	1	brrelex12i	⊢ ( 𝐴 ≼ 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
3		brdom2g	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
4	2 3	syl	⊢ ( 𝐴 ≼ 𝐵 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
5	4	ibi	⊢ ( 𝐴 ≼ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )