Theorem rescom

Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998)

	Ref	Expression
Assertion	rescom	$$⊢{\left({A\upharpoonright }_B\right)\upharpoonright }_C={\left({A\upharpoonright }_C\right)\upharpoonright }_B$$

Proof

Step	Hyp	Ref	Expression
1		incom	$$⊢B\cap C=C\cap B$$
2	1	reseq2i	$$⊢{A\upharpoonright }_{\left(B\cap C\right)}={A\upharpoonright }_{\left(C\cap B\right)}$$
3		resres	$$⊢{\left({A\upharpoonright }_B\right)\upharpoonright }_C={A\upharpoonright }_{\left(B\cap C\right)}$$
4		resres	$$⊢{\left({A\upharpoonright }_C\right)\upharpoonright }_B={A\upharpoonright }_{\left(C\cap B\right)}$$
5	2 3 4	3eqtr4i	$$⊢{\left({A\upharpoonright }_B\right)\upharpoonright }_C={\left({A\upharpoonright }_C\right)\upharpoonright }_B$$