Theorem suprclrnmpt

Description: Closure of the indexed supremum of a nonempty bounded set of reals. Range of a function in maps-to notation can be used, to express an indexed supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	suprclrnmpt.x	|- F/ x ph
	suprclrnmpt.n	|- ( ph -> A =/= (/) )
	suprclrnmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR )
	suprclrnmpt.y	|- ( ph -> E. y e. RR A. x e. A B <_ y )
Assertion	suprclrnmpt	|- ( ph -> sup ( ran ( x e. A |-> B ) , RR , < ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		suprclrnmpt.x	|- F/ x ph
2		suprclrnmpt.n	|- ( ph -> A =/= (/) )
3		suprclrnmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR )
4		suprclrnmpt.y	|- ( ph -> E. y e. RR A. x e. A B <_ y )
5		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
6	1 5 3	rnmptssd	|- ( ph -> ran ( x e. A |-> B ) C_ RR )
7	1 3 5 2	rnmptn0	|- ( ph -> ran ( x e. A |-> B ) =/= (/) )
8	1 4	rnmptbdd	|- ( ph -> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y )
9	6 7 8	suprcld	|- ( ph -> sup ( ran ( x e. A |-> B ) , RR , < ) e. RR )