Theorem rncossdmcoss

Description: The range of cosets is the domain of them (this should be rncoss but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019)

	Ref	Expression
Assertion	rncossdmcoss	|- ran ,~ R = dom ,~ R

Proof

Step	Hyp	Ref	Expression
1		brcosscnvcoss	|- ( ( y e. _V /\ x e. _V ) -> ( y ,~ R x <-> x ,~ R y ) )
2	1	el2v	|- ( y ,~ R x <-> x ,~ R y )
3	2	exbii	|- ( E. y y ,~ R x <-> E. y x ,~ R y )
4	3	abbii	|- { x | E. y y ,~ R x } = { x | E. y x ,~ R y }
5		dfrn2	|- ran ,~ R = { x | E. y y ,~ R x }
6		df-dm	|- dom ,~ R = { x | E. y x ,~ R y }
7	4 5 6	3eqtr4i	|- ran ,~ R = dom ,~ R