Theorem reseq1

Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994)

	Ref	Expression
Assertion	reseq1	$$⊢A=B\to {A\upharpoonright }_C={B\upharpoonright }_C$$

Proof

Step	Hyp	Ref	Expression
1		ineq1	$$⊢A=B\to A\cap \left(C\times \mathrm{V}\right)=B\cap \left(C\times \mathrm{V}\right)$$
2		df-res	$$⊢{A\upharpoonright }_C=A\cap \left(C\times \mathrm{V}\right)$$
3		df-res	$$⊢{B\upharpoonright }_C=B\cap \left(C\times \mathrm{V}\right)$$
4	1 2 3	3eqtr4g	$$⊢A=B\to {A\upharpoonright }_C={B\upharpoonright }_C$$