Theorem brparts2

Description: Binary partitions relation. (Contributed by Peter Mazsa, 30-Dec-2021)

	Ref	Expression
Assertion	brparts2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Parts 𝐴 ↔ ( 𝑅 ∈ Disjs ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brparts	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑅 Parts 𝐴 ↔ ( 𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴 ) ) )
2	1	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Parts 𝐴 ↔ ( 𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴 ) ) )
3		brdmqss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
4	3	anbi2d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ( 𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴 ) ↔ ( 𝑅 ∈ Disjs ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ) )
5	2 4	bitrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Parts 𝐴 ↔ ( 𝑅 ∈ Disjs ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ) )