Metamath Proof Explorer

Theorem filunibas

Description: Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion filunibas 
|- ( F e. ( Fil ` X ) -> U. F = X )

Proof

Step Hyp Ref Expression
1 filsspw 
 |-  ( F e. ( Fil ` X ) -> F C_ ~P X )
2 sspwuni 
 |-  ( F C_ ~P X <-> U. F C_ X )
3 1 2 sylib 
 |-  ( F e. ( Fil ` X ) -> U. F C_ X )
4 filtop 
 |-  ( F e. ( Fil ` X ) -> X e. F )
5 unissel 
 |-  ( ( U. F C_ X /\ X e. F ) -> U. F = X )
6 3 4 5 syl2anc 
 |-  ( F e. ( Fil ` X ) -> U. F = X )