Metamath Proof Explorer


Theorem fiuni

Description: The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009) (Revised by Mario Carneiro, 24-Nov-2013)

Ref Expression
Assertion fiuni ( 𝐴𝑉 𝐴 = ( fi ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ssfii ( 𝐴𝑉𝐴 ⊆ ( fi ‘ 𝐴 ) )
2 1 unissd ( 𝐴𝑉 𝐴 ( fi ‘ 𝐴 ) )
3 fipwuni ( fi ‘ 𝐴 ) ⊆ 𝒫 𝐴
4 3 unissi ( fi ‘ 𝐴 ) ⊆ 𝒫 𝐴
5 unipw 𝒫 𝐴 = 𝐴
6 4 5 sseqtri ( fi ‘ 𝐴 ) ⊆ 𝐴
7 6 a1i ( 𝐴𝑉 ( fi ‘ 𝐴 ) ⊆ 𝐴 )
8 2 7 eqssd ( 𝐴𝑉 𝐴 = ( fi ‘ 𝐴 ) )