Theorem relecxrn

Description: The ( R |X. S ) -coset of a set is a relation. (Contributed by Peter Mazsa, 15-Oct-2020)

	Ref	Expression
Assertion	relecxrn	$$⊢A\in V\to \mathrm{Rel}\left[A\right]R⋉S$$

Proof

Step	Hyp	Ref	Expression
1		relopab	$$⊢\mathrm{Rel}\left\{⟨y,z⟩|\left(ARy\wedge ASz\right)\right\}$$
2		ecxrn	$$⊢A\in V\to \left[A\right]R⋉S=\left\{⟨y,z⟩|\left(ARy\wedge ASz\right)\right\}$$
3	2	releqd	$$⊢A\in V\to \left(\mathrm{Rel}\left[A\right]R⋉S\leftrightarrow \mathrm{Rel}\left\{⟨y,z⟩|\left(ARy\wedge ASz\right)\right\}\right)$$
4	1 3	mpbiri	$$⊢A\in V\to \mathrm{Rel}\left[A\right]R⋉S$$