Theorem sibfrn

Description: A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018)

	Ref	Expression
Hypotheses	sitgval.b	|- B = ( Base ` W )
	sitgval.j	|- J = ( TopOpen ` W )
	sitgval.s	|- S = ( sigaGen ` J )
	sitgval.0	|- .0. = ( 0g ` W )
	sitgval.x	|- .x. = ( .s ` W )
	sitgval.h	|- H = ( RRHom ` ( Scalar ` W ) )
	sitgval.1	|- ( ph -> W e. V )
	sitgval.2	|- ( ph -> M e. U. ran measures )
	sibfmbl.1	|- ( ph -> F e. dom ( W sitg M ) )
Assertion	sibfrn	|- ( ph -> ran F e. Fin )

Proof

Step	Hyp	Ref	Expression
1		sitgval.b	|- B = ( Base ` W )
2		sitgval.j	|- J = ( TopOpen ` W )
3		sitgval.s	|- S = ( sigaGen ` J )
4		sitgval.0	|- .0. = ( 0g ` W )
5		sitgval.x	|- .x. = ( .s ` W )
6		sitgval.h	|- H = ( RRHom ` ( Scalar ` W ) )
7		sitgval.1	|- ( ph -> W e. V )
8		sitgval.2	|- ( ph -> M e. U. ran measures )
9		sibfmbl.1	|- ( ph -> F e. dom ( W sitg M ) )
10	1 2 3 4 5 6 7 8	issibf	|- ( ph -> ( F e. dom ( W sitg M ) <-> ( F e. ( dom M MblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )
11	9 10	mpbid	|- ( ph -> ( F e. ( dom M MblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
12	11	simp2d	|- ( ph -> ran F e. Fin )