Theorem eldmcoss

Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019)

	Ref	Expression
Assertion	eldmcoss	$$⊢A\in V\to \left(A\in \mathrm{dom}\wr R\leftrightarrow \exists uuRA\right)$$

Proof

Step	Hyp	Ref	Expression
1		dmcoss3	$$⊢\mathrm{dom}\wr R=\mathrm{dom}{R}^{-1}$$
2	1	eleq2i	$$⊢A\in \mathrm{dom}\wr R\leftrightarrow A\in \mathrm{dom}{R}^{-1}$$
3		eldmcnv	$$⊢A\in V\to \left(A\in \mathrm{dom}{R}^{-1}\leftrightarrow \exists uuRA\right)$$
4	2 3	bitrid	$$⊢A\in V\to \left(A\in \mathrm{dom}\wr R\leftrightarrow \exists uuRA\right)$$