Theorem imanonrel

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

	Ref	Expression
Assertion	imanonrel	$$⊢\left(A\setminus {A^{-1}}^{-1}\right)\left[B\right]=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		df-ima	$$⊢\left(A\setminus {A^{-1}}^{-1}\right)\left[B\right]=\mathrm{ran}\left({\left(A\setminus {A^{-1}}^{-1}\right)\upharpoonright }_B\right)$$
2		resnonrel	$$⊢{\left(A\setminus {A^{-1}}^{-1}\right)\upharpoonright }_B=\varnothing$$
3	2	rneqi	$$⊢\mathrm{ran}\left({\left(A\setminus {A^{-1}}^{-1}\right)\upharpoonright }_B\right)=\mathrm{ran}\varnothing$$
4		rn0	$$⊢\mathrm{ran}\varnothing =\varnothing$$
5	1 3 4	3eqtri	$$⊢\left(A\setminus {A^{-1}}^{-1}\right)\left[B\right]=\varnothing$$