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 ) )