Theorem brdom

Description: Dominance relation. (Contributed by NM, 15-Jun-1998)

	Ref	Expression
Hypothesis	bren.1	⊢ 𝐵 ∈ V
Assertion	brdom	⊢ ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bren.1	⊢ 𝐵 ∈ V
2		brdomg	⊢ ( 𝐵 ∈ V → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )