Theorem unidmqseq

Description: The union of the domain quotient of a relation is equal to the class A if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019) (Revised by Peter Mazsa, 28-Dec-2021)

	Ref	Expression
Assertion	unidmqseq	$$⊢R\in V\to \left(\mathrm{Rel}R\to \left(\bigcup \left(\mathrm{dom}R/R\right)=A\leftrightarrow \mathrm{ran}R=A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		unidmqs	$$⊢R\in V\to \left(\mathrm{Rel}R\to \bigcup \left(\mathrm{dom}R/R\right)=\mathrm{ran}R\right)$$
2	1	imp	$$⊢\left(R\in V\wedge \mathrm{Rel}R\right)\to \bigcup \left(\mathrm{dom}R/R\right)=\mathrm{ran}R$$
3	2	eqeq1d	$$⊢\left(R\in V\wedge \mathrm{Rel}R\right)\to \left(\bigcup \left(\mathrm{dom}R/R\right)=A\leftrightarrow \mathrm{ran}R=A\right)$$
4	3	ex	$$⊢R\in V\to \left(\mathrm{Rel}R\to \left(\bigcup \left(\mathrm{dom}R/R\right)=A\leftrightarrow \mathrm{ran}R=A\right)\right)$$