Theorem dmxrncnvepres2

Description: Domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 29-Jan-2026)

	Ref	Expression
Assertion	dmxrncnvepres2	$$⊢\mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\cap \left(\mathrm{dom}R\setminus \left\{\varnothing \right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		dmres	$$⊢\mathrm{dom}\left({R\upharpoonright }_A\right)=A\cap \mathrm{dom}R$$
2	1	difeq1i	$$⊢\mathrm{dom}\left({R\upharpoonright }_A\right)\setminus \left\{\varnothing \right\}=\left(A\cap \mathrm{dom}R\right)\setminus \left\{\varnothing \right\}$$
3		dmxrncnvepres	$$⊢\mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=\mathrm{dom}\left({R\upharpoonright }_A\right)\setminus \left\{\varnothing \right\}$$
4		indif2	$$⊢A\cap \left(\mathrm{dom}R\setminus \left\{\varnothing \right\}\right)=\left(A\cap \mathrm{dom}R\right)\setminus \left\{\varnothing \right\}$$
5	2 3 4	3eqtr4i	$$⊢\mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\cap \left(\mathrm{dom}R\setminus \left\{\varnothing \right\}\right)$$