Theorem ressbasss2

Description: The base set of a restriction to A is a subset of A . (Contributed by SN, 10-Jan-2025)

	Ref	Expression
Hypothesis	ressbasss2.r	⊢ 𝑅 = ( 𝑊 ↾s 𝐴 )
Assertion	ressbasss2	⊢ ( Base ‘ 𝑅 ) ⊆ 𝐴

Proof

Step	Hyp	Ref	Expression
1		ressbasss2.r	⊢ 𝑅 = ( 𝑊 ↾s 𝐴 )
2		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
3	1 2	ressbasssg	⊢ ( Base ‘ 𝑅 ) ⊆ ( 𝐴 ∩ ( Base ‘ 𝑊 ) )
4		inss1	⊢ ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⊆ 𝐴
5	3 4	sstri	⊢ ( Base ‘ 𝑅 ) ⊆ 𝐴