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Metamath Proof Explorer

Definition df-part

Description: Define the partition predicate (read: A is a partition by R ). Alternative definition is dfpart2 . The binary partition and the partition predicate are the same if A and R are sets, cf. brpartspart . (Contributed by Peter Mazsa, 12-Aug-2021)

		Ref	Expression
	Assertion	df-part	⊢ ( 𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴 ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cR	⊢ 𝑅
1		cA	⊢ 𝐴
2	1 0	wpart	⊢ 𝑅 Part 𝐴
3	0	wdisjALTV	⊢ Disj 𝑅
4	1 0	wdmqs	⊢ 𝑅 DomainQs 𝐴
5	3 4	wa	⊢ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴 )
6	2 5	wb	⊢ ( 𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴 ) )