Theorem supxrre3rnmpt

Description: The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	supxrre3rnmpt.x	|- F/ x ph
	supxrre3rnmpt.a	|- ( ph -> A =/= (/) )
	supxrre3rnmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR )
Assertion	supxrre3rnmpt	|- ( ph -> ( sup ( ran ( x e. A |-> B ) , RR* , < ) e. RR <-> E. y e. RR A. x e. A B <_ y ) )

Proof

Step	Hyp	Ref	Expression
1		supxrre3rnmpt.x	|- F/ x ph
2		supxrre3rnmpt.a	|- ( ph -> A =/= (/) )
3		supxrre3rnmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR )
4		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
5	1 4 3	rnmptssd	|- ( ph -> ran ( x e. A |-> B ) C_ RR )
6	1 3 4 2	rnmptn0	|- ( ph -> ran ( x e. A |-> B ) =/= (/) )
7		supxrre3	|- ( ( ran ( x e. A |-> B ) C_ RR /\ ran ( x e. A |-> B ) =/= (/) ) -> ( sup ( ran ( x e. A |-> B ) , RR* , < ) e. RR <-> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y ) )
8	5 6 7	syl2anc	|- ( ph -> ( sup ( ran ( x e. A |-> B ) , RR* , < ) e. RR <-> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y ) )
9	1 3	rnmptbd	|- ( ph -> ( E. y e. RR A. x e. A B <_ y <-> E. y e. RR A. z e. ran ( x e. A |-> B ) z <_ y ) )
10	8 9	bitr4d	|- ( ph -> ( sup ( ran ( x e. A |-> B ) , RR* , < ) e. RR <-> E. y e. RR A. x e. A B <_ y ) )