Theorem dmxrncnvepres

Description: Domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 23-Nov-2025)

	Ref	Expression
Assertion	dmxrncnvepres	$$⊢\mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=\mathrm{dom}\left({R\upharpoonright }_A\right)\setminus \left\{\varnothing \right\}$$

Proof

Step	Hyp	Ref	Expression
1		xrnres	$$⊢{R⋉{\mathrm{E}}^{-1}\upharpoonright }_A=\left({R\upharpoonright }_A\right)⋉{\mathrm{E}}^{-1}$$
2		xrnres2	$$⊢{R⋉{\mathrm{E}}^{-1}\upharpoonright }_A=R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
3	1 2	eqtr3i	$$⊢\left({R\upharpoonright }_A\right)⋉{\mathrm{E}}^{-1}=R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
4	3	dmeqi	$$⊢\mathrm{dom}\left({R\upharpoonright }_A\right)⋉{\mathrm{E}}^{-1}=\mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
5		dmxrncnvep	$$⊢\mathrm{dom}\left({R\upharpoonright }_A\right)⋉{\mathrm{E}}^{-1}=\mathrm{dom}\left({R\upharpoonright }_A\right)\setminus \left\{\varnothing \right\}$$
6	4 5	eqtr3i	$$⊢\mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=\mathrm{dom}\left({R\upharpoonright }_A\right)\setminus \left\{\varnothing \right\}$$