This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dmqseqim

Description: If the domain quotient of a relation is equal to the class A , then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021)

		Ref	Expression
	Assertion	dmqseqim	$⊢ R \in V \to (Rel R \to (dom R / R = A \to ran R = ⋃ A))$

Proof

Step	Hyp	Ref	Expression
1		unieq	$⊢ dom R / R = A \to ⋃ (dom R / R) = ⋃ A$
2		unidmqseq	$⊢ R \in V \to (Rel R \to (⋃ (dom R / R) = ⋃ A \leftrightarrow ran R = ⋃ A))$
3	2	imp	$⊢ (R \in V \land Rel R) \to (⋃ (dom R / R) = ⋃ A \leftrightarrow ran R = ⋃ A)$
4	1 3	imbitrid	$⊢ (R \in V \land Rel R) \to (dom R / R = A \to ran R = ⋃ A)$
5	4	ex	$⊢ R \in V \to (Rel R \to (dom R / R = A \to ran R = ⋃ A))$