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Metamath Proof Explorer

Theorem mptss

Description: Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020)

		Ref	Expression
	Assertion	mptss	$⊢ A \subseteq B \to (x \in A ⟼ C) \subseteq (x \in B ⟼ C)$

Step	Hyp	Ref	Expression
1		resmpt	$⊢ A \subseteq B \to {(x \in B ⟼ C) ↾}_{A} = (x \in A ⟼ C)$
2		resss	$⊢ {(x \in B ⟼ C) ↾}_{A} \subseteq (x \in B ⟼ C)$
3	1 2	eqsstrrdi	$⊢ A \subseteq B \to (x \in A ⟼ C) \subseteq (x \in B ⟼ C)$