Metamath Proof Explorer


Theorem ressbasss

Description: The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 30-Apr-2015) (Proof shortened by SN, 25-Feb-2025)

Ref Expression
Hypotheses ressbas.r 𝑅 = ( 𝑊s 𝐴 )
ressbas.b 𝐵 = ( Base ‘ 𝑊 )
Assertion ressbasss ( Base ‘ 𝑅 ) ⊆ 𝐵

Proof

Step Hyp Ref Expression
1 ressbas.r 𝑅 = ( 𝑊s 𝐴 )
2 ressbas.b 𝐵 = ( Base ‘ 𝑊 )
3 1 2 ressbasssg ( Base ‘ 𝑅 ) ⊆ ( 𝐴𝐵 )
4 inss2 ( 𝐴𝐵 ) ⊆ 𝐵
5 3 4 sstri ( Base ‘ 𝑅 ) ⊆ 𝐵