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Metamath Proof Explorer

Theorem resdifcom

Description: Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021)

		Ref	Expression
	Assertion	resdifcom	$⊢ ({A ↾}_{B}) ∖ C = {(A ∖ C) ↾}_{B}$

Step	Hyp	Ref	Expression
1		indif1	$⊢ (A ∖ C) \cap (B \times V) = (A \cap (B \times V)) ∖ C$
2		df-res	$⊢ {(A ∖ C) ↾}_{B} = (A ∖ C) \cap (B \times V)$
3		df-res	$⊢ {A ↾}_{B} = A \cap (B \times V)$
4	3	difeq1i	$⊢ ({A ↾}_{B}) ∖ C = (A \cap (B \times V)) ∖ C$
5	1 2 4	3eqtr4ri	$⊢ ({A ↾}_{B}) ∖ C = {(A ∖ C) ↾}_{B}$