Theorem ressucdifsn

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class. (Contributed by Peter Mazsa, 20-Sep-2024)

	Ref	Expression
Assertion	ressucdifsn	$$⊢\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$

Proof

Step	Hyp	Ref	Expression
1		df-suc	$$⊢\mathrm{suc}A=A\cup \left\{A\right\}$$
2	1	reseq2i	$$⊢{R\upharpoonright }_{\mathrm{suc}A}={R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}$$
3	2	difeq1i	$$⊢\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)=\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)$$
4		ressucdifsn2	$$⊢\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$
5	3 4	eqtri	$$⊢\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$