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Metamath Proof Explorer

Theorem suprubrnmpt

Description: A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021)

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

Proof

Step Hyp Ref Expression
1 suprubrnmpt.x 
 |-  F/ x ph
2 suprubrnmpt.b 
 |-  ( ( ph /\ x e. A ) -> B e. RR )
3 suprubrnmpt.e 
 |-  ( ph -> E. y e. RR A. x e. A B <_ y )
4 eqid 
 |-  ( x e. A |-> B ) = ( x e. A |-> B )
5 1 4 2 rnmptssd 
 |-  ( ph -> ran ( x e. A |-> B ) C_ RR )
6 5 adantr 
 |-  ( ( ph /\ x e. A ) -> ran ( x e. A |-> B ) C_ RR )
7 simpr 
 |-  ( ( ph /\ x e. A ) -> x e. A )
8 4 elrnmpt1 
 |-  ( ( x e. A /\ B e. RR ) -> B e. ran ( x e. A |-> B ) )
9 7 2 8 syl2anc 
 |-  ( ( ph /\ x e. A ) -> B e. ran ( x e. A |-> B ) )
10 9 ne0d 
 |-  ( ( ph /\ x e. A ) -> ran ( x e. A |-> B ) =/= (/) )
11 1 3 rnmptbdd 
 |-  ( ph -> E. y e. RR A. w e. ran ( x e. A |-> B ) w <_ y )
12 11 adantr 
 |-  ( ( ph /\ x e. A ) -> E. y e. RR A. w e. ran ( x e. A |-> B ) w <_ y )
13 6 10 12 9 suprubd 
 |-  ( ( ph /\ x e. A ) -> B <_ sup ( ran ( x e. A |-> B ) , RR , < ) )