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 =/= (/) ) |
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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 ) |
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 ) |