Metamath Proof Explorer

Theorem dfpart2

Description: Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion dfpart2 ⊢ R Part A ↔ Disj R ∧ dom ⁡ R / R = A

Proof

Step Hyp Ref Expression
1 df-part ⊢ R Part A ↔ Disj R ∧ R DomainQs A
2 df-dmqs ⊢ R DomainQs A ↔ dom ⁡ R / R = A
3 2 anbi2i ⊢ Disj R ∧ R DomainQs A ↔ Disj R ∧ dom ⁡ R / R = A
4 1 3 bitri ⊢ R Part A ↔ Disj R ∧ dom ⁡ R / R = A