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Theorem bndth 21436
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 21248 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
53, 4eqeltri 2527 . . . . . . 7  |-  K  e.  (TopOn `  RR )
65toponunii 19411 . . . . . 6  |-  RR  =  U. K
72, 6cnf 19725 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
81, 7syl 16 . . . 4  |-  ( ph  ->  F : X --> RR )
9 frn 5727 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . 3  |-  ( ph  ->  ran  F  C_  RR )
11 imassrn 5338 . . . . . 6  |-  ( (,) " ( { -oo }  X.  RR ) ) 
C_  ran  (,)
12 retopbas 21245 . . . . . . . 8  |-  ran  (,)  e. 
TopBases
13 bastg 19445 . . . . . . . 8  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
1412, 13ax-mp 5 . . . . . . 7  |-  ran  (,)  C_  ( topGen `  ran  (,) )
1514, 3sseqtr4i 3522 . . . . . 6  |-  ran  (,)  C_  K
1611, 15sstri 3498 . . . . 5  |-  ( (,) " ( { -oo }  X.  RR ) ) 
C_  K
17 retop 21246 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  Top
183, 17eqeltri 2527 . . . . . . 7  |-  K  e. 
Top
1918elexi 3105 . . . . . 6  |-  K  e. 
_V
2019elpw2 4601 . . . . 5  |-  ( ( (,) " ( { -oo }  X.  RR ) )  e.  ~P K 
<->  ( (,) " ( { -oo }  X.  RR ) )  C_  K
)
2116, 20mpbir 209 . . . 4  |-  ( (,) " ( { -oo }  X.  RR ) )  e.  ~P K
22 bndth.3 . . . . . 6  |-  ( ph  ->  J  e.  Comp )
23 rncmp 19874 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
2422, 1, 23syl2anc 661 . . . . 5  |-  ( ph  ->  ( Kt  ran  F )  e. 
Comp )
256cmpsub 19878 . . . . . 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 663 . . . . 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 210 . . . 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 4242 . . . . . . . 8  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  U. u  =  U. ( (,) " ( { -oo }  X.  RR ) ) )
2911unissi 4257 . . . . . . . . . 10  |-  U. ( (,) " ( { -oo }  X.  RR ) ) 
C_  U. ran  (,)
30 unirnioo 11635 . . . . . . . . . 10  |-  RR  =  U. ran  (,)
3129, 30sseqtr4i 3522 . . . . . . . . 9  |-  U. ( (,) " ( { -oo }  X.  RR ) ) 
C_  RR
32 id 22 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  RR )
33 ltp1 10387 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <  ( x  +  1 ) )
34 ressxr 9640 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
35 peano2re 9756 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
3634, 35sseldi 3487 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR* )
37 elioomnf 11630 . . . . . . . . . . . . 13  |-  ( ( x  +  1 )  e.  RR*  ->  ( x  e.  ( -oo (,) ( x  +  1
) )  <->  ( x  e.  RR  /\  x  < 
( x  +  1 ) ) ) )
3836, 37syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  ( -oo (,) ( x  +  1 ) )  <->  ( x  e.  RR  /\  x  < 
( x  +  1 ) ) ) )
3932, 33, 38mpbir2and 922 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  ( -oo (,) (
x  +  1 ) ) )
40 df-ov 6284 . . . . . . . . . . . 12  |-  ( -oo (,) ( x  +  1 ) )  =  ( (,) `  <. -oo , 
( x  +  1 ) >. )
41 mnfxr 11334 . . . . . . . . . . . . . . . 16  |- -oo  e.  RR*
4241elexi 3105 . . . . . . . . . . . . . . 15  |- -oo  e.  _V
4342snid 4042 . . . . . . . . . . . . . 14  |- -oo  e.  { -oo }
44 opelxpi 5021 . . . . . . . . . . . . . 14  |-  ( ( -oo  e.  { -oo }  /\  ( x  + 
1 )  e.  RR )  ->  <. -oo ,  ( x  +  1 ) >.  e.  ( { -oo }  X.  RR ) )
4543, 35, 44sylancr 663 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  <. -oo , 
( x  +  1 ) >.  e.  ( { -oo }  X.  RR ) )
46 ioof 11633 . . . . . . . . . . . . . . 15  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
47 ffun 5723 . . . . . . . . . . . . . . 15  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
4846, 47ax-mp 5 . . . . . . . . . . . . . 14  |-  Fun  (,)
49 snssi 4159 . . . . . . . . . . . . . . . . 17  |-  ( -oo  e.  RR*  ->  { -oo }  C_ 
RR* )
5041, 49ax-mp 5 . . . . . . . . . . . . . . . 16  |-  { -oo } 
C_  RR*
51 xpss12 5098 . . . . . . . . . . . . . . . 16  |-  ( ( { -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
5250, 34, 51mp2an 672 . . . . . . . . . . . . . . 15  |-  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* )
5346fdmi 5726 . . . . . . . . . . . . . . 15  |-  dom  (,)  =  ( RR*  X.  RR* )
5452, 53sseqtr4i 3522 . . . . . . . . . . . . . 14  |-  ( { -oo }  X.  RR )  C_  dom  (,)
55 funfvima2 6133 . . . . . . . . . . . . . 14  |-  ( ( Fun  (,)  /\  ( { -oo }  X.  RR )  C_  dom  (,) )  ->  ( <. -oo ,  ( x  +  1 )
>.  e.  ( { -oo }  X.  RR )  -> 
( (,) `  <. -oo ,  ( x  + 
1 ) >. )  e.  ( (,) " ( { -oo }  X.  RR ) ) ) )
5648, 54, 55mp2an 672 . . . . . . . . . . . . 13  |-  ( <. -oo ,  ( x  + 
1 ) >.  e.  ( { -oo }  X.  RR )  ->  ( (,) `  <. -oo ,  ( x  +  1 ) >.
)  e.  ( (,) " ( { -oo }  X.  RR ) ) )
5745, 56syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  ( (,) `  <. -oo ,  ( x  +  1 )
>. )  e.  ( (,) " ( { -oo }  X.  RR ) ) )
5840, 57syl5eqel 2535 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  ( -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( { -oo }  X.  RR ) ) )
59 elunii 4239 . . . . . . . . . . 11  |-  ( ( x  e.  ( -oo (,) ( x  +  1 ) )  /\  ( -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( { -oo }  X.  RR ) ) )  ->  x  e.  U. ( (,) " ( { -oo }  X.  RR ) ) )
6039, 58, 59syl2anc 661 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  U. ( (,) " ( { -oo }  X.  RR ) ) )
6160ssriv 3493 . . . . . . . . 9  |-  RR  C_  U. ( (,) " ( { -oo }  X.  RR ) )
6231, 61eqssi 3505 . . . . . . . 8  |-  U. ( (,) " ( { -oo }  X.  RR ) )  =  RR
6328, 62syl6eq 2500 . . . . . . 7  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  U. u  =  RR )
6463sseq2d 3517 . . . . . 6  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  ( ran  F  C_  U. u  <->  ran 
F  C_  RR )
)
65 pweq 4000 . . . . . . . 8  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  ~P u  =  ~P ( (,) " ( { -oo }  X.  RR ) ) )
6665ineq1d 3684 . . . . . . 7  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  ( ~P u  i^i  Fin )  =  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )
6766rexeqdv 3047 . . . . . 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
) )
6864, 67imbi12d 320 . . . . 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
) ) )
6968rspcv 3192 . . . 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
) ) )
7021, 27, 69mpsyl 63 . . 3  |-  ( ph  ->  ( ran  F  C_  RR  ->  E. v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) ran  F  C_  U. v
) )
7110, 70mpd 15 . 2  |-  ( ph  ->  E. v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) ran  F  C_  U. v
)
72 simpr 461 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )
73 elin 3672 . . . . . . 7  |-  ( v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  <->  ( v  e.  ~P ( (,) " ( { -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7472, 73sylib 196 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  (
v  e.  ~P ( (,) " ( { -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7574adantrr 716 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  (
v  e.  ~P ( (,) " ( { -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7675simprd 463 . . . 4  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  v  e.  Fin )
7774simpld 459 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ~P ( (,) " ( { -oo }  X.  RR ) ) )
7877elpwid 4007 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  C_  ( (,) " ( { -oo }  X.  RR ) ) )
7950sseli 3485 . . . . . . . . . . . 12  |-  ( u  e.  { -oo }  ->  u  e.  RR* )
8079adantr 465 . . . . . . . . . . 11  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  u  e.  RR* )
8134sseli 3485 . . . . . . . . . . . 12  |-  ( w  e.  RR  ->  w  e.  RR* )
8281adantl 466 . . . . . . . . . . 11  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  w  e.  RR* )
83 mnflt 11344 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  -> -oo  <  w )
84 xrltnle 9656 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  w  e.  RR* )  ->  ( -oo  <  w  <->  -.  w  <_ -oo ) )
8541, 81, 84sylancr 663 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  ->  ( -oo  <  w  <->  -.  w  <_ -oo ) )
8683, 85mpbid 210 . . . . . . . . . . . . . 14  |-  ( w  e.  RR  ->  -.  w  <_ -oo )
8786adantl 466 . . . . . . . . . . . . 13  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  -.  w  <_ -oo )
88 elsni 4039 . . . . . . . . . . . . . . 15  |-  ( u  e.  { -oo }  ->  u  = -oo )
8988adantr 465 . . . . . . . . . . . . . 14  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  u  = -oo )
9089breq2d 4449 . . . . . . . . . . . . 13  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( w  <_  u 
<->  w  <_ -oo )
)
9187, 90mtbird 301 . . . . . . . . . . . 12  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  -.  w  <_  u )
92 ioo0 11565 . . . . . . . . . . . . . 14  |-  ( ( u  e.  RR*  /\  w  e.  RR* )  ->  (
( u (,) w
)  =  (/)  <->  w  <_  u ) )
9379, 81, 92syl2an 477 . . . . . . . . . . . . 13  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =  (/) 
<->  w  <_  u )
)
9493necon3abid 2689 . . . . . . . . . . . 12  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =/=  (/) 
<->  -.  w  <_  u
) )
9591, 94mpbird 232 . . . . . . . . . . 11  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( u (,) w )  =/=  (/) )
96 df-ioo 11544 . . . . . . . . . . . 12  |-  (,)  =  ( y  e.  RR* ,  z  e.  RR*  |->  { v  e.  RR*  |  (
y  <  v  /\  v  <  z ) } )
97 idd 24 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <  w ) )
98 xrltle 11366 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <_  w ) )
99 idd 24 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <  x ) )
100 xrltle 11366 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <_  x ) )
10196, 97, 98, 99, 100ixxub 11561 . . . . . . . . . . 11  |-  ( ( u  e.  RR*  /\  w  e.  RR*  /\  ( u (,) w )  =/=  (/) )  ->  sup (
( u (,) w
) ,  RR* ,  <  )  =  w )
10280, 82, 95, 101syl3anc 1229 . . . . . . . . . 10  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  =  w )
103 simpr 461 . . . . . . . . . 10  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  w  e.  RR )
104102, 103eqeltrd 2531 . . . . . . . . 9  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  e.  RR )
105104rgen2 2868 . . . . . . . 8  |-  A. u  e.  { -oo } A. w  e.  RR  sup ( ( u (,) w ) ,  RR* ,  <  )  e.  RR
106 fveq2 5856 . . . . . . . . . . . 12  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( (,) `  <. u ,  w >. ) )
107 df-ov 6284 . . . . . . . . . . . 12  |-  ( u (,) w )  =  ( (,) `  <. u ,  w >. )
108106, 107syl6eqr 2502 . . . . . . . . . . 11  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( u (,) w ) )
109108supeq1d 7908 . . . . . . . . . 10  |-  ( z  =  <. u ,  w >.  ->  sup ( ( (,) `  z ) ,  RR* ,  <  )  =  sup ( ( u (,) w ) ,  RR* ,  <  ) )
110109eleq1d 2512 . . . . . . . . 9  |-  ( z  =  <. u ,  w >.  ->  ( sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR  <->  sup (
( u (,) w
) ,  RR* ,  <  )  e.  RR ) )
111110ralxp 5134 . . . . . . . 8  |-  ( 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 )
112105, 111mpbir 209 . . . . . . 7  |-  A. z  e.  ( { -oo }  X.  RR ) sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR
113 ffn 5721 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
11446, 113ax-mp 5 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
115 supeq1 7907 . . . . . . . . . 10  |-  ( w  =  ( (,) `  z
)  ->  sup (
w ,  RR* ,  <  )  =  sup ( ( (,) `  z ) ,  RR* ,  <  )
)
116115eleq1d 2512 . . . . . . . . 9  |-  ( w  =  ( (,) `  z
)  ->  ( sup ( w ,  RR* ,  <  )  e.  RR  <->  sup ( ( (,) `  z
) ,  RR* ,  <  )  e.  RR ) )
117116ralima 6137 . . . . . . . 8  |-  ( ( (,)  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 ) )
118114, 52, 117mp2an 672 . . . . . . 7  |-  ( A. w  e.  ( (,) " ( { -oo }  X.  RR ) ) sup ( w ,  RR* ,  <  )  e.  RR  <->  A. z  e.  ( { -oo }  X.  RR ) sup ( ( (,) `  z ) ,  RR* ,  <  )  e.  RR )
119112, 118mpbir 209 . . . . . 6  |-  A. w  e.  ( (,) " ( { -oo }  X.  RR ) ) sup (
w ,  RR* ,  <  )  e.  RR
120 ssralv 3549 . . . . . 6  |-  ( 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 ) )
12178, 119, 120mpisyl 18 . . . . 5  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
122121adantrr 716 . . . 4  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
123 fimaxre3 10499 . . . 4  |-  ( ( v  e.  Fin  /\  A. w  e.  v  sup ( w ,  RR* ,  <  )  e.  RR )  ->  E. x  e.  RR  A. w  e.  v  sup ( w ,  RR* ,  <  )  <_  x
)
12476, 122, 123syl2anc 661 . . 3  |-  ( (
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 )
125 simplrr 762 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  ran  F  C_  U. v
)
126125sselda 3489 . . . . . . 7  |-  ( ( ( ( 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 )
127 eluni2 4238 . . . . . . . 8  |-  ( z  e.  U. v  <->  E. w  e.  v  z  e.  w )
128 r19.29r 2979 . . . . . . . . . 10  |-  ( ( 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 ) )
129 sspwuni 4401 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (,) " ( { -oo }  X.  RR ) )  C_  ~P RR 
<-> 
U. ( (,) " ( { -oo }  X.  RR ) )  C_  RR )
13031, 129mpbir 209 . . . . . . . . . . . . . . . . . 18  |-  ( (,) " ( { -oo }  X.  RR ) ) 
C_  ~P RR
131783ad2ant1 1018 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) ) )
132 simp2r 1024 . . . . . . . . . . . . . . . . . . 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.  v )
133131, 132sseldd 3490 . . . . . . . . . . . . . . . . . 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  e.  ( (,) " ( { -oo }  X.  RR ) ) )
134130, 133sseldi 3487 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 )
135134elpwid 4007 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 )
136 simp3l 1025 . . . . . . . . . . . . . . . 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  e.  w )
137135, 136sseldd 3490 . . . . . . . . . . . . . . 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  e.  RR )
138121r19.21bi 2812 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  /\  w  e.  v )  ->  sup (
w ,  RR* ,  <  )  e.  RR )
139138adantrl 715 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v ) )  ->  sup ( w ,  RR* ,  <  )  e.  RR )
1401393adant3 1017 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 )
141 simp2l 1023 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 )
142135, 34syl6ss 3501 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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* )
143 supxrub 11527 . . . . . . . . . . . . . . . 16  |-  ( ( w  C_  RR*  /\  z  e.  w )  ->  z  <_  sup ( w , 
RR* ,  <  ) )
144142, 136, 143syl2anc 661 . . . . . . . . . . . . . . 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  <_  sup ( w , 
RR* ,  <  ) )
145 simp3r 1026 . . . . . . . . . . . . . . 15  |-  ( ( ( 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
)
146137, 140, 141, 144, 145letrd 9742 . . . . . . . . . . . . . 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 )
1471463expia 1199 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
)
148147anassrs 648 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) )
149148rexlimdva 2935 . . . . . . . . . . 11  |-  ( ( ( 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 )
)
150149adantlrr 720 . . . . . . . . . 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  /\  sup (
w ,  RR* ,  <  )  <_  x )  -> 
z  <_  x )
)
151128, 150syl5 32 . . . . . . . . 9  |-  ( ( ( 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 ) )
152151expdimp 437 . . . . . . . 8  |-  ( ( ( ( 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 ) )
153127, 152sylan2b 475 . . . . . . 7  |-  ( ( ( ( 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 )
)
154126, 153syldan 470 . . . . . 6  |-  ( ( ( ( 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 )
)
155154ralrimdva 2861 . . . . 5  |-  ( ( ( 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 ) )
156 ffn 5721 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
1578, 156syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
158157ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  F  Fn  X )
159 breq1 4440 . . . . . . 7  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
160159ralrn 6019 . . . . . 6  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
161158, 160syl 16 . . . . 5  |-  ( ( ( 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 ) )
162155, 161sylibd 214 . . . 4  |-  ( ( ( 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
) )
163162reximdva 2918 . . 3  |-  ( (
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 ) )
164124, 163mpd 15 . 2  |-  ( (
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
)
16571, 164rexlimddv 2939 1  |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794    i^i cin 3460    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   {csn 4014   <.cop 4020   U.cuni 4234   class class class wbr 4437    X. cxp 4987   dom cdm 4989   ran crn 4990   "cima 4992   Fun wfun 5572    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   Fincfn 7518   supcsup 7902   RRcr 9494   1c1 9496    + caddc 9498   -oocmnf 9629   RR*cxr 9630    < clt 9631    <_ cle 9632   (,)cioo 11540   ↾t crest 14800   topGenctg 14817   Topctop 19372  TopOnctopon 19373   TopBasesctb 19376    Cn ccn 19703   Compccmp 19864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-q 11194  df-ioo 11544  df-rest 14802  df-topgen 14823  df-top 19377  df-bases 19379  df-topon 19380  df-cn 19706  df-cmp 19865
This theorem is referenced by:  evth  21437
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