Theorem brpartspart

Description: Binary partition and the partition predicate are the same if A and R are sets. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	brpartspart	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eldisjsdisj	⊢ ( 𝑅 ∈ 𝑊 → ( 𝑅 ∈ Disjs ↔ Disj 𝑅 ) )
2	1	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 ∈ Disjs ↔ Disj 𝑅 ) )
3		brdmqssqs	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴 ) )
4	2 3	anbi12d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ( 𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴 ) ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴 ) ) )
5		brparts	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑅 Parts 𝐴 ↔ ( 𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴 ) ) )
6	5	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Parts 𝐴 ↔ ( 𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴 ) ) )
7		df-part	⊢ ( 𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴 ) )
8	7	a1i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴 ) ) )
9	4 6 8	3bitr4d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 Parts 𝐴 ↔ 𝑅 Part 𝐴 ) )