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 e. V /\ R e. W ) -> ( R Parts A <-> R Part A ) )

Proof

Step	Hyp	Ref	Expression
1		eldisjsdisj	|- ( R e. W -> ( R e. Disjs <-> Disj R ) )
2	1	adantl	|- ( ( A e. V /\ R e. W ) -> ( R e. Disjs <-> Disj R ) )
3		brdmqssqs	|- ( ( A e. V /\ R e. W ) -> ( R DomainQss A <-> R DomainQs A ) )
4	2 3	anbi12d	|- ( ( A e. V /\ R e. W ) -> ( ( R e. Disjs /\ R DomainQss A ) <-> ( Disj R /\ R DomainQs A ) ) )
5		brparts	|- ( A e. V -> ( R Parts A <-> ( R e. Disjs /\ R DomainQss A ) ) )
6	5	adantr	|- ( ( A e. V /\ R e. W ) -> ( R Parts A <-> ( R e. Disjs /\ R DomainQss A ) ) )
7		df-part	|- ( R Part A <-> ( Disj R /\ R DomainQs A ) )
8	7	a1i	|- ( ( A e. V /\ R e. W ) -> ( R Part A <-> ( Disj R /\ R DomainQs A ) ) )
9	4 6 8	3bitr4d	|- ( ( A e. V /\ R e. W ) -> ( R Parts A <-> R Part A ) )