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Metamath Proof Explorer

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 ) )