Theorem resabs1

Description: Absorption law for restriction. Exercise 17 of TakeutiZaring p. 25. (Contributed by NM, 9-Aug-1994)

	Ref	Expression
Assertion	resabs1	$$⊢B\subseteq C\to {\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_B$$

Proof

Step	Hyp	Ref	Expression
1		resres	$$⊢{\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_{\left(C\cap B\right)}$$
2		sseqin2	$$⊢B\subseteq C\leftrightarrow C\cap B=B$$
3		reseq2	$$⊢C\cap B=B\to {A\upharpoonright }_{\left(C\cap B\right)}={A\upharpoonright }_B$$
4	2 3	sylbi	$$⊢B\subseteq C\to {A\upharpoonright }_{\left(C\cap B\right)}={A\upharpoonright }_B$$
5	1 4	eqtrid	$$⊢B\subseteq C\to {\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_B$$