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Metamath Proof Explorer

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 ) )