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Metamath Proof Explorer

Theorem bndss

Description: A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion bndss 
|- ( ( M e. ( Bnd ` X ) /\ S C_ X ) -> ( M |` ( S X. S ) ) e. ( Bnd ` S ) )

Proof

Step Hyp Ref Expression
1 metres2 
 |-  ( ( M e. ( Met ` X ) /\ S C_ X ) -> ( M |` ( S X. S ) ) e. ( Met ` S ) )
2 1 adantlr 
 |-  ( ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ S C_ X ) -> ( M |` ( S X. S ) ) e. ( Met ` S ) )
3 ssel2 
 |-  ( ( S C_ X /\ x e. S ) -> x e. X )
4 3 ancoms 
 |-  ( ( x e. S /\ S C_ X ) -> x e. X )
5 oveq1 
 |-  ( y = x -> ( y ( ball ` M ) r ) = ( x ( ball ` M ) r ) )
6 5 eqeq2d 
 |-  ( y = x -> ( X = ( y ( ball ` M ) r ) <-> X = ( x ( ball ` M ) r ) ) )
7 6 rexbidv 
 |-  ( y = x -> ( E. r e. RR+ X = ( y ( ball ` M ) r ) <-> E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
8 7 rspcva 
 |-  ( ( x e. X /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) -> E. r e. RR+ X = ( x ( ball ` M ) r ) )
9 4 8 sylan 
 |-  ( ( ( x e. S /\ S C_ X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) -> E. r e. RR+ X = ( x ( ball ` M ) r ) )
10 9 adantlll 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) -> E. r e. RR+ X = ( x ( ball ` M ) r ) )
11 dfss 
 |-  ( S C_ X <-> S = ( S i^i X ) )
12 11 biimpi 
 |-  ( S C_ X -> S = ( S i^i X ) )
13 incom 
 |-  ( S i^i X ) = ( X i^i S )
14 12 13 eqtrdi 
 |-  ( S C_ X -> S = ( X i^i S ) )
15 ineq1 
 |-  ( X = ( x ( ball ` M ) r ) -> ( X i^i S ) = ( ( x ( ball ` M ) r ) i^i S ) )
16 14 15 sylan9eq 
 |-  ( ( S C_ X /\ X = ( x ( ball ` M ) r ) ) -> S = ( ( x ( ball ` M ) r ) i^i S ) )
17 16 adantll 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ X = ( x ( ball ` M ) r ) ) -> S = ( ( x ( ball ` M ) r ) i^i S ) )
18 17 adantlr 
 |-  ( ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ r e. RR+ ) /\ X = ( x ( ball ` M ) r ) ) -> S = ( ( x ( ball ` M ) r ) i^i S ) )
19 eqid 
 |-  ( M |` ( S X. S ) ) = ( M |` ( S X. S ) )
20 19 blssp 
 |-  ( ( ( M e. ( Met ` X ) /\ S C_ X ) /\ ( x e. S /\ r e. RR+ ) ) -> ( x ( ball ` ( M |` ( S X. S ) ) ) r ) = ( ( x ( ball ` M ) r ) i^i S ) )
21 20 an4s 
 |-  ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ ( S C_ X /\ r e. RR+ ) ) -> ( x ( ball ` ( M |` ( S X. S ) ) ) r ) = ( ( x ( ball ` M ) r ) i^i S ) )
22 21 anassrs 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ r e. RR+ ) -> ( x ( ball ` ( M |` ( S X. S ) ) ) r ) = ( ( x ( ball ` M ) r ) i^i S ) )
23 22 adantr 
 |-  ( ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ r e. RR+ ) /\ X = ( x ( ball ` M ) r ) ) -> ( x ( ball ` ( M |` ( S X. S ) ) ) r ) = ( ( x ( ball ` M ) r ) i^i S ) )
24 18 23 eqtr4d 
 |-  ( ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ r e. RR+ ) /\ X = ( x ( ball ` M ) r ) ) -> S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
25 24 ex 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ r e. RR+ ) -> ( X = ( x ( ball ` M ) r ) -> S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) ) )
26 25 reximdva 
 |-  ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) -> ( E. r e. RR+ X = ( x ( ball ` M ) r ) -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) ) )
27 26 imp 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ E. r e. RR+ X = ( x ( ball ` M ) r ) ) -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
28 10 27 syldan 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ S C_ X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
29 28 an32s 
 |-  ( ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ S C_ X ) -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
30 29 ex 
 |-  ( ( ( M e. ( Met ` X ) /\ x e. S ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) -> ( S C_ X -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) ) )
31 30 an32s 
 |-  ( ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ x e. S ) -> ( S C_ X -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) ) )
32 31 imp 
 |-  ( ( ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ x e. S ) /\ S C_ X ) -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
33 32 an32s 
 |-  ( ( ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ S C_ X ) /\ x e. S ) -> E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
34 33 ralrimiva 
 |-  ( ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ S C_ X ) -> A. x e. S E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) )
35 2 34 jca 
 |-  ( ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ S C_ X ) -> ( ( M |` ( S X. S ) ) e. ( Met ` S ) /\ A. x e. S E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) ) )
36 isbnd 
 |-  ( M e. ( Bnd ` X ) <-> ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) )
37 36 anbi1i 
 |-  ( ( M e. ( Bnd ` X ) /\ S C_ X ) <-> ( ( M e. ( Met ` X ) /\ A. y e. X E. r e. RR+ X = ( y ( ball ` M ) r ) ) /\ S C_ X ) )
38 isbnd 
 |-  ( ( M |` ( S X. S ) ) e. ( Bnd ` S ) <-> ( ( M |` ( S X. S ) ) e. ( Met ` S ) /\ A. x e. S E. r e. RR+ S = ( x ( ball ` ( M |` ( S X. S ) ) ) r ) ) )
39 35 37 38 3imtr4i 
 |-  ( ( M e. ( Bnd ` X ) /\ S C_ X ) -> ( M |` ( S X. S ) ) e. ( Bnd ` S ) )