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Theorem bndth 18456
Description: The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to  -u F.) (Contributed by Mario Carneiro, 12-Aug-2014.)
Hypotheses
Ref Expression
bndth.1  |-  X  = 
U. J
bndth.2  |-  K  =  ( topGen `  ran  (,) )
bndth.3  |-  ( ph  ->  J  e.  Comp )
bndth.4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
bndth  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
Distinct variable groups:    x, y, F    y, K    ph, x, y   
x, X, y    x, J, y
Allowed substitution hint:    K( x)

Proof of Theorem bndth
Dummy variables  v  u  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bndth.4 . . . . 5  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2 bndth.1 . . . . . 6  |-  X  = 
U. J
3 bndth.2 . . . . . . . 8  |-  K  =  ( topGen `  ran  (,) )
4 retopon 18272 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
53, 4eqeltri 2353 . . . . . . 7  |-  K  e.  (TopOn `  RR )
65toponunii 16670 . . . . . 6  |-  RR  =  U. K
72, 6cnf 16976 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
81, 7syl 15 . . . 4  |-  ( ph  ->  F : X --> RR )
9 frn 5395 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
108, 9syl 15 . . 3  |-  ( ph  ->  ran  F  C_  RR )
11 imassrn 5025 . . . . . 6  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  ran  (,)
12 retopbas 18269 . . . . . . . 8  |-  ran  (,)  e. 
TopBases
13 bastg 16704 . . . . . . . 8  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
1412, 13ax-mp 8 . . . . . . 7  |-  ran  (,)  C_  ( topGen `  ran  (,) )
1514, 3sseqtr4i 3211 . . . . . 6  |-  ran  (,)  C_  K
1611, 15sstri 3188 . . . . 5  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  K
17 retop 18270 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  Top
183, 17eqeltri 2353 . . . . . . 7  |-  K  e. 
Top
1918elexi 2797 . . . . . 6  |-  K  e. 
_V
2019elpw2 4175 . . . . 5  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  e.  ~P K 
<->  ( (,) " ( {  -oo }  X.  RR ) )  C_  K
)
2116, 20mpbir 200 . . . 4  |-  ( (,) " ( {  -oo }  X.  RR ) )  e.  ~P K
22 bndth.3 . . . . . 6  |-  ( ph  ->  J  e.  Comp )
23 rncmp 17123 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
2422, 1, 23syl2anc 642 . . . . 5  |-  ( ph  ->  ( Kt  ran  F )  e. 
Comp )
256cmpsub 17127 . . . . . 6  |-  ( ( K  e.  Top  /\  ran  F  C_  RR )  ->  ( ( Kt  ran  F
)  e.  Comp  <->  A. u  e.  ~P  K ( ran 
F  C_  U. u  ->  E. v  e.  ( ~P u  i^i  Fin ) ran  F  C_  U. v
) ) )
2618, 10, 25sylancr 644 . . . . 5  |-  ( ph  ->  ( ( Kt  ran  F
)  e.  Comp  <->  A. u  e.  ~P  K ( ran 
F  C_  U. u  ->  E. v  e.  ( ~P u  i^i  Fin ) ran  F  C_  U. v
) ) )
2724, 26mpbid 201 . . . 4  |-  ( ph  ->  A. u  e.  ~P  K ( ran  F  C_ 
U. u  ->  E. v  e.  ( ~P u  i^i 
Fin ) ran  F  C_ 
U. v ) )
28 unieq 3836 . . . . . . . 8  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  U. u  =  U. ( (,) " ( {  -oo }  X.  RR ) ) )
29 uniss 3848 . . . . . . . . . . 11  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  C_  ran  (,) 
->  U. ( (,) " ( {  -oo }  X.  RR ) )  C_  U. ran  (,) )
3011, 29ax-mp 8 . . . . . . . . . 10  |-  U. ( (,) " ( {  -oo }  X.  RR ) ) 
C_  U. ran  (,)
31 unirnioo 10743 . . . . . . . . . 10  |-  RR  =  U. ran  (,)
3230, 31sseqtr4i 3211 . . . . . . . . 9  |-  U. ( (,) " ( {  -oo }  X.  RR ) ) 
C_  RR
33 id 19 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  RR )
34 ltp1 9594 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <  ( x  +  1 ) )
35 ressxr 8876 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
36 peano2re 8985 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
3735, 36sseldi 3178 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR* )
38 elioomnf 10738 . . . . . . . . . . . . 13  |-  ( ( x  +  1 )  e.  RR*  ->  ( x  e.  (  -oo (,) ( x  +  1
) )  <->  ( x  e.  RR  /\  x  < 
( x  +  1 ) ) ) )
3937, 38syl 15 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  (  -oo (,) ( x  +  1 ) )  <->  ( x  e.  RR  /\  x  < 
( x  +  1 ) ) ) )
4033, 34, 39mpbir2and 888 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  (  -oo (,) (
x  +  1 ) ) )
41 df-ov 5861 . . . . . . . . . . . 12  |-  (  -oo (,) ( x  +  1 ) )  =  ( (,) `  <.  -oo , 
( x  +  1 ) >. )
42 mnfxr 10456 . . . . . . . . . . . . . . . 16  |-  -oo  e.  RR*
4342elexi 2797 . . . . . . . . . . . . . . 15  |-  -oo  e.  _V
4443snid 3667 . . . . . . . . . . . . . 14  |-  -oo  e.  { 
-oo }
45 opelxpi 4721 . . . . . . . . . . . . . 14  |-  ( ( 
-oo  e.  {  -oo }  /\  ( x  +  1 )  e.  RR )  ->  <.  -oo ,  ( x  +  1 ) >.  e.  ( {  -oo }  X.  RR ) )
4644, 36, 45sylancr 644 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  <.  -oo , 
( x  +  1 ) >.  e.  ( {  -oo }  X.  RR ) )
47 ioof 10741 . . . . . . . . . . . . . . 15  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
48 ffun 5391 . . . . . . . . . . . . . . 15  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
4947, 48ax-mp 8 . . . . . . . . . . . . . 14  |-  Fun  (,)
50 snssi 3759 . . . . . . . . . . . . . . . . 17  |-  (  -oo  e.  RR*  ->  {  -oo }  C_ 
RR* )
5142, 50ax-mp 8 . . . . . . . . . . . . . . . 16  |-  {  -oo } 
C_  RR*
52 xpss12 4792 . . . . . . . . . . . . . . . 16  |-  ( ( {  -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( {  -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
5351, 35, 52mp2an 653 . . . . . . . . . . . . . . 15  |-  ( { 
-oo }  X.  RR )  C_  ( RR*  X.  RR* )
5447fdmi 5394 . . . . . . . . . . . . . . 15  |-  dom  (,)  =  ( RR*  X.  RR* )
5553, 54sseqtr4i 3211 . . . . . . . . . . . . . 14  |-  ( { 
-oo }  X.  RR )  C_  dom  (,)
56 funfvima2 5754 . . . . . . . . . . . . . 14  |-  ( ( Fun  (,)  /\  ( {  -oo }  X.  RR )  C_  dom  (,) )  ->  ( <.  -oo ,  ( x  +  1 )
>.  e.  ( {  -oo }  X.  RR )  -> 
( (,) `  <.  -oo
,  ( x  + 
1 ) >. )  e.  ( (,) " ( {  -oo }  X.  RR ) ) ) )
5749, 55, 56mp2an 653 . . . . . . . . . . . . 13  |-  ( <.  -oo ,  ( x  + 
1 ) >.  e.  ( {  -oo }  X.  RR )  ->  ( (,) `  <.  -oo ,  ( x  +  1 ) >.
)  e.  ( (,) " ( {  -oo }  X.  RR ) ) )
5846, 57syl 15 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  ( (,) `  <.  -oo ,  ( x  +  1 )
>. )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )
5941, 58syl5eqel 2367 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  (  -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )
60 elunii 3832 . . . . . . . . . . 11  |-  ( ( x  e.  (  -oo (,) ( x  +  1 ) )  /\  (  -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )  ->  x  e.  U. ( (,) " ( {  -oo }  X.  RR ) ) )
6140, 59, 60syl2anc 642 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  U. ( (,) " ( {  -oo }  X.  RR ) ) )
6261ssriv 3184 . . . . . . . . 9  |-  RR  C_  U. ( (,) " ( {  -oo }  X.  RR ) )
6332, 62eqssi 3195 . . . . . . . 8  |-  U. ( (,) " ( {  -oo }  X.  RR ) )  =  RR
6428, 63syl6eq 2331 . . . . . . 7  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  U. u  =  RR )
6564sseq2d 3206 . . . . . 6  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( ran  F  C_  U. u  <->  ran 
F  C_  RR )
)
66 pweq 3628 . . . . . . . 8  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ~P u  =  ~P ( (,) " ( {  -oo }  X.  RR ) ) )
6766ineq1d 3369 . . . . . . 7  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( ~P u  i^i  Fin )  =  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )
6867rexeqdv 2743 . . . . . 6  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( E. v  e.  ( ~P u  i^i  Fin ) ran  F  C_  U. v  <->  E. v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) ran  F  C_  U. v
) )
6965, 68imbi12d 311 . . . . 5  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  (
( ran  F  C_  U. u  ->  E. v  e.  ( ~P u  i^i  Fin ) ran  F  C_  U. v
)  <->  ( ran  F  C_  RR  ->  E. v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) ran 
F  C_  U. v
) ) )
7069rspcv 2880 . . . 4  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  e.  ~P K  ->  ( A. u  e.  ~P  K ( ran 
F  C_  U. u  ->  E. v  e.  ( ~P u  i^i  Fin ) ran  F  C_  U. v
)  ->  ( ran  F 
C_  RR  ->  E. v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) ran 
F  C_  U. v
) ) )
7121, 27, 70mpsyl 59 . . 3  |-  ( ph  ->  ( ran  F  C_  RR  ->  E. v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) ran  F  C_  U. v
) )
7210, 71mpd 14 . 2  |-  ( ph  ->  E. v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) ran  F  C_  U. v
)
73 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )
74 elin 3358 . . . . . . . . 9  |-  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  <->  ( v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7573, 74sylib 188 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  (
v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7675adantrr 697 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  (
v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7776simprd 449 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  v  e.  Fin )
7875simpld 445 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) ) )
79 elpwi 3633 . . . . . . . . 9  |-  ( v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  ->  v  C_  ( (,) " ( {  -oo }  X.  RR ) ) )
8078, 79syl 15 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  C_  ( (,) " ( {  -oo }  X.  RR ) ) )
8151sseli 3176 . . . . . . . . . . . . . 14  |-  ( u  e.  {  -oo }  ->  u  e.  RR* )
8281adantr 451 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  u  e.  RR* )
8335sseli 3176 . . . . . . . . . . . . . 14  |-  ( w  e.  RR  ->  w  e.  RR* )
8483adantl 452 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  w  e.  RR* )
85 mnflt 10464 . . . . . . . . . . . . . . . . 17  |-  ( w  e.  RR  ->  -oo  <  w )
86 xrltnle 8891 . . . . . . . . . . . . . . . . . 18  |-  ( ( 
-oo  e.  RR*  /\  w  e.  RR* )  ->  (  -oo  <  w  <->  -.  w  <_  -oo ) )
8742, 83, 86sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( w  e.  RR  ->  (  -oo  <  w  <->  -.  w  <_  -oo ) )
8885, 87mpbid 201 . . . . . . . . . . . . . . . 16  |-  ( w  e.  RR  ->  -.  w  <_  -oo )
8988adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  -.  w  <_  -oo )
90 elsni 3664 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  {  -oo }  ->  u  =  -oo )
9190adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  u  =  -oo )
9291breq2d 4035 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( w  <_  u 
<->  w  <_  -oo ) )
9389, 92mtbird 292 . . . . . . . . . . . . . 14  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  -.  w  <_  u )
94 ioo0 10681 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  RR*  /\  w  e.  RR* )  ->  (
( u (,) w
)  =  (/)  <->  w  <_  u ) )
9581, 83, 94syl2an 463 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =  (/) 
<->  w  <_  u )
)
9695necon3abid 2479 . . . . . . . . . . . . . 14  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =/=  (/) 
<->  -.  w  <_  u
) )
9793, 96mpbird 223 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( u (,) w )  =/=  (/) )
98 df-ioo 10660 . . . . . . . . . . . . . 14  |-  (,)  =  ( y  e.  RR* ,  z  e.  RR*  |->  { v  e.  RR*  |  (
y  <  v  /\  v  <  z ) } )
99 idd 21 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <  w ) )
100 xrltle 10483 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <_  w ) )
101 idd 21 . . . . . . . . . . . . . 14  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <  x ) )
102 xrltle 10483 . . . . . . . . . . . . . 14  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <_  x ) )
10398, 99, 100, 101, 102ixxub 10677 . . . . . . . . . . . . 13  |-  ( ( u  e.  RR*  /\  w  e.  RR*  /\  ( u (,) w )  =/=  (/) )  ->  sup (
( u (,) w
) ,  RR* ,  <  )  =  w )
10482, 84, 97, 103syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  =  w )
105 simpr 447 . . . . . . . . . . . 12  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  w  e.  RR )
106104, 105eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  e.  RR )
107106rgen2 2639 . . . . . . . . . 10  |-  A. u  e.  {  -oo } A. w  e.  RR  sup ( ( u (,) w ) ,  RR* ,  <  )  e.  RR
108 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( (,) `  <. u ,  w >. ) )
109 df-ov 5861 . . . . . . . . . . . . . 14  |-  ( u (,) w )  =  ( (,) `  <. u ,  w >. )
110108, 109syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( u (,) w ) )
111110supeq1d 7199 . . . . . . . . . . . 12  |-  ( z  =  <. u ,  w >.  ->  sup ( ( (,) `  z ) ,  RR* ,  <  )  =  sup ( ( u (,) w ) ,  RR* ,  <  ) )
112111eleq1d 2349 . . . . . . . . . . 11  |-  ( z  =  <. u ,  w >.  ->  ( sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR  <->  sup (
( u (,) w
) ,  RR* ,  <  )  e.  RR ) )
113112ralxp 4827 . . . . . . . . . 10  |-  ( A. z  e.  ( {  -oo }  X.  RR ) sup ( ( (,) `  z ) ,  RR* ,  <  )  e.  RR  <->  A. u  e.  {  -oo } A. w  e.  RR  sup ( ( u (,) w ) ,  RR* ,  <  )  e.  RR )
114107, 113mpbir 200 . . . . . . . . 9  |-  A. z  e.  ( {  -oo }  X.  RR ) sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR
115 ffn 5389 . . . . . . . . . . 11  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
11647, 115ax-mp 8 . . . . . . . . . 10  |-  (,)  Fn  ( RR*  X.  RR* )
117 supeq1 7198 . . . . . . . . . . . 12  |-  ( w  =  ( (,) `  z
)  ->  sup (
w ,  RR* ,  <  )  =  sup ( ( (,) `  z ) ,  RR* ,  <  )
)
118117eleq1d 2349 . . . . . . . . . . 11  |-  ( w  =  ( (,) `  z
)  ->  ( sup ( w ,  RR* ,  <  )  e.  RR  <->  sup ( ( (,) `  z
) ,  RR* ,  <  )  e.  RR ) )
119118ralima 5758 . . . . . . . . . 10  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  ( {  -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )  ->  ( A. w  e.  ( (,) " ( {  -oo }  X.  RR ) ) sup ( w , 
RR* ,  <  )  e.  RR  <->  A. z  e.  ( {  -oo }  X.  RR ) sup ( ( (,) `  z ) ,  RR* ,  <  )  e.  RR ) )
120116, 53, 119mp2an 653 . . . . . . . . 9  |-  ( A. w  e.  ( (,) " ( {  -oo }  X.  RR ) ) sup ( w ,  RR* ,  <  )  e.  RR  <->  A. z  e.  ( { 
-oo }  X.  RR ) sup ( ( (,) `  z ) ,  RR* ,  <  )  e.  RR )
121114, 120mpbir 200 . . . . . . . 8  |-  A. w  e.  ( (,) " ( {  -oo }  X.  RR ) ) sup (
w ,  RR* ,  <  )  e.  RR
122 ssralv 3237 . . . . . . . 8  |-  ( v 
C_  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( A. w  e.  ( (,) " ( {  -oo }  X.  RR ) ) sup ( w , 
RR* ,  <  )  e.  RR  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR ) )
12380, 121, 122ee10 1366 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
124123adantrr 697 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
125 fimaxre3 9703 . . . . . 6  |-  ( ( v  e.  Fin  /\  A. w  e.  v  sup ( w ,  RR* ,  <  )  e.  RR )  ->  E. x  e.  RR  A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x
)
12677, 124, 125syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  E. x  e.  RR  A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x )
127 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ran  F  C_  U. v
)
128127sselda 3180 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  /\  z  e.  ran  F )  ->  z  e.  U. v )
129 eluni2 3831 . . . . . . . . . 10  |-  ( z  e.  U. v  <->  E. w  e.  v  z  e.  w )
130 r19.29r 2684 . . . . . . . . . . . 12  |-  ( ( E. w  e.  v  z  e.  w  /\  A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x
)  ->  E. w  e.  v  ( z  e.  w  /\  sup (
w ,  RR* ,  <  )  <_  x ) )
131 sspwuni 3987 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  C_  ~P RR 
<-> 
U. ( (,) " ( {  -oo }  X.  RR ) )  C_  RR )
13232, 131mpbir 200 . . . . . . . . . . . . . . . . . . . 20  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  ~P RR
133803ad2ant1 976 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  v  C_  ( (,) " ( {  -oo }  X.  RR ) ) )
134 simp2r 982 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  w  e.  v )
135133, 134sseldd 3181 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  w  e.  ( (,) " ( {  -oo }  X.  RR ) ) )
136132, 135sseldi 3178 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  w  e.  ~P RR )
137 elpwi 3633 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e.  ~P RR  ->  w 
C_  RR )
138136, 137syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  w  C_  RR )
139 simp3l 983 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  z  e.  w )
140138, 139sseldd 3181 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  z  e.  RR )
141123r19.21bi 2641 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  w  e.  v )  ->  sup (
w ,  RR* ,  <  )  e.  RR )
142141adantrl 696 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v ) )  ->  sup ( w ,  RR* ,  <  )  e.  RR )
1431423adant3 975 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  sup ( w ,  RR* ,  <  )  e.  RR )
144 simp2l 981 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  x  e.  RR )
145138, 35syl6ss 3191 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  w  C_ 
RR* )
146 supxrub 10643 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  C_  RR*  /\  z  e.  w )  ->  z  <_  sup ( w , 
RR* ,  <  ) )
147145, 139, 146syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  z  <_  sup ( w , 
RR* ,  <  ) )
148 simp3r 984 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  sup ( w ,  RR* ,  <  )  <_  x
)
149140, 143, 144, 147, 148letrd 8973 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v )  /\  (
z  e.  w  /\  sup ( w ,  RR* ,  <  )  <_  x
) )  ->  z  <_  x )
1501493expia 1153 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v ) )  -> 
( ( z  e.  w  /\  sup (
w ,  RR* ,  <  )  <_  x )  -> 
z  <_  x )
)
151150anassrs 629 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( { 
-oo }  X.  RR ) )  i^i  Fin ) )  /\  x  e.  RR )  /\  w  e.  v )  ->  (
( z  e.  w  /\  sup ( w , 
RR* ,  <  )  <_  x )  ->  z  <_  x ) )
152151rexlimdva 2667 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  x  e.  RR )  ->  ( E. w  e.  v  ( z  e.  w  /\  sup (
w ,  RR* ,  <  )  <_  x )  -> 
z  <_  x )
)
153152adantlrr 701 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ( E. w  e.  v  ( z  e.  w  /\  sup (
w ,  RR* ,  <  )  <_  x )  -> 
z  <_  x )
)
154130, 153syl5 28 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ( ( E. w  e.  v  z  e.  w  /\  A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x )  ->  z  <_  x ) )
155154expdimp 426 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  /\  E. w  e.  v  z  e.  w )  ->  ( A. w  e.  v  sup (
w ,  RR* ,  <  )  <_  x  ->  z  <_  x ) )
156129, 155sylan2b 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  /\  z  e.  U. v
)  ->  ( A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x  ->  z  <_  x )
)
157128, 156syldan 456 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  /\  z  e.  ran  F )  ->  ( A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x  ->  z  <_  x )
)
158157ralrimdva 2633 . . . . . . 7  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ( A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x  ->  A. z  e.  ran  F  z  <_  x ) )
159 ffn 5389 . . . . . . . . . 10  |-  ( F : X --> RR  ->  F  Fn  X )
1608, 159syl 15 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
161160ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  F  Fn  X )
162 breq1 4026 . . . . . . . . 9  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
163162ralrn 5668 . . . . . . . 8  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
164161, 163syl 15 . . . . . . 7  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ( A. z  e. 
ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x ) )
165158, 164sylibd 205 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ( A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x  ->  A. y  e.  X  ( F `  y )  <_  x
) )
166165reximdva 2655 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  ( E. x  e.  RR  A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x ) )
167126, 166mpd 14 . . . 4  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x
)
168167expr 598 . . 3  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  ( ran  F  C_  U. v  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x ) )
169168rexlimdva 2667 . 2  |-  ( ph  ->  ( E. v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) ran 
F  C_  U. v  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x ) )
17072, 169mpd 14 1  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023    X. cxp 4687   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   ↾t crest 13325   topGenctg 13342   Topctop 16631  TopOnctopon 16632   TopBasesctb 16635    Cn ccn 16954   Compccmp 17113
This theorem is referenced by:  evth  18457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-ioo 10660  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cmp 17114
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