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Metamath Proof Explorer

Theorem fmptssfisupp

Description: The restriction of a mapping function has finite support if that function has finite support. (Contributed by Thierry Arnoux, 21-Jan-2024)

	Ref	Expression
Hypotheses	fmptssfisupp.1	$⊢ φ \to {finSupp}_{Z} ((x \in A ⟼ B))$
	fmptssfisupp.2	$⊢ φ \to C \subseteq A$
	fmptssfisupp.3	$⊢ φ \to Z \in V$
Assertion	fmptssfisupp	$⊢ φ \to {finSupp}_{Z} ((x \in C ⟼ B))$

Proof

Step	Hyp	Ref	Expression
1		fmptssfisupp.1	$⊢ φ \to {finSupp}_{Z} ((x \in A ⟼ B))$
2		fmptssfisupp.2	$⊢ φ \to C \subseteq A$
3		fmptssfisupp.3	$⊢ φ \to Z \in V$
4	2	resmptd	$⊢ φ \to {(x \in A ⟼ B) ↾}_{C} = (x \in C ⟼ B)$
5	1 3	fsuppres	$⊢ φ \to {finSupp}_{Z} (({(x \in A ⟼ B) ↾}_{C}))$
6	4 5	eqbrtrrd	$⊢ φ \to {finSupp}_{Z} ((x \in C ⟼ B))$