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 ) )