Theorem uffinfix

Description: An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

	Ref	Expression
Assertion	uffinfix	|- ( ( F e. ( UFil ` X ) /\ S e. F /\ S e. Fin ) -> E. x e. X F = { y e. ~P X | x e. y } )

Proof

Step	Hyp	Ref	Expression
1		ufilfil	|- ( F e. ( UFil ` X ) -> F e. ( Fil ` X ) )
2		filfinnfr	|- ( ( F e. ( Fil ` X ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )
3	1 2	syl3an1	|- ( ( F e. ( UFil ` X ) /\ S e. F /\ S e. Fin ) -> |^| F =/= (/) )
4		uffix2	|- ( F e. ( UFil ` X ) -> ( |^| F =/= (/) <-> E. x e. X F = { y e. ~P X | x e. y } ) )
5	4	3ad2ant1	|- ( ( F e. ( UFil ` X ) /\ S e. F /\ S e. Fin ) -> ( |^| F =/= (/) <-> E. x e. X F = { y e. ~P X | x e. y } ) )
6	3 5	mpbid	|- ( ( F e. ( UFil ` X ) /\ S e. F /\ S e. Fin ) -> E. x e. X F = { y e. ~P X | x e. y } )