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Metamath Proof Explorer

Theorem partimeq

Description: Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq . (Contributed by Peter Mazsa, 25-Dec-2024)

		Ref	Expression
	Assertion	partimeq	$⊢ R \in V \to (R Part A \to \sim A = ≀ R)$

Proof

Step	Hyp	Ref	Expression
1		cossex	$⊢ R \in V \to ≀ R \in V$
2		partim	$⊢ R Part A \to ≀ R ErALTV A$
3		erimeq	$⊢ ≀ R \in V \to (≀ R ErALTV A \to \sim A = ≀ R)$
4	1 2 3	syl2im	$⊢ R \in V \to (R Part A \to \sim A = ≀ R)$