Theorem brco

Description: Binary relation on a composition. (Contributed by NM, 21-Sep-2004) (Revised by Mario Carneiro, 24-Feb-2015)

	Ref	Expression
Hypotheses	opelco.1	⊢ 𝐴 ∈ V
	opelco.2	⊢ 𝐵 ∈ V
Assertion	brco	⊢ ( 𝐴 ( 𝐶 ∘ 𝐷 ) 𝐵 ↔ ∃ 𝑥 ( 𝐴 𝐷 𝑥 ∧ 𝑥 𝐶 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		opelco.1	⊢ 𝐴 ∈ V
2		opelco.2	⊢ 𝐵 ∈ V
3		brcog	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ( 𝐶 ∘ 𝐷 ) 𝐵 ↔ ∃ 𝑥 ( 𝐴 𝐷 𝑥 ∧ 𝑥 𝐶 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ( 𝐶 ∘ 𝐷 ) 𝐵 ↔ ∃ 𝑥 ( 𝐴 𝐷 𝑥 ∧ 𝑥 𝐶 𝐵 ) )