Theorem dfcoss4

Description: Alternate definition of the class of cosets by R (see the comment of df-coss ). (Contributed by Peter Mazsa, 12-Jul-2021)

	Ref	Expression
Assertion	dfcoss4	$$⊢\wr R=\mathrm{ran}R⋉R$$

Proof

Step	Hyp	Ref	Expression
1		df-coss	$$⊢\wr R=\left\{⟨x,y⟩|\exists u\left(uRx\wedge uRy\right)\right\}$$
2		rnxrn	$$⊢\mathrm{ran}R⋉R=\left\{⟨x,y⟩|\exists u\left(uRx\wedge uRy\right)\right\}$$
3	1 2	eqtr4i	$$⊢\wr R=\mathrm{ran}R⋉R$$