Theorem dmnonrel

Description: The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020)

	Ref	Expression
Assertion	dmnonrel	$$⊢\mathrm{dom}\left(A\setminus {A^{-1}}^{-1}\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		dfdm4	$$⊢\mathrm{dom}\left(A\setminus {A^{-1}}^{-1}\right)=\mathrm{ran}{\left(A\setminus {A^{-1}}^{-1}\right)}^{-1}$$
2		cnvnonrel	$$⊢{\left(A\setminus {A^{-1}}^{-1}\right)}^{-1}=\varnothing$$
3	2	rneqi	$$⊢\mathrm{ran}{\left(A\setminus {A^{-1}}^{-1}\right)}^{-1}=\mathrm{ran}\varnothing$$
4		rn0	$$⊢\mathrm{ran}\varnothing =\varnothing$$
5	1 3 4	3eqtri	$$⊢\mathrm{dom}\left(A\setminus {A^{-1}}^{-1}\right)=\varnothing$$