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Metamath Proof Explorer

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 ) )