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Theorem bndth 22064
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 21862 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
53, 4eqeltri 2545 . . . . . . 7  |-  K  e.  (TopOn `  RR )
65toponunii 20024 . . . . . 6  |-  RR  =  U. K
72, 6cnf 20339 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
81, 7syl 17 . . . 4  |-  ( ph  ->  F : X --> RR )
9 frn 5747 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
108, 9syl 17 . . 3  |-  ( ph  ->  ran  F  C_  RR )
11 imassrn 5185 . . . . . 6  |-  ( (,) " ( { -oo }  X.  RR ) ) 
C_  ran  (,)
12 retopbas 21859 . . . . . . . 8  |-  ran  (,)  e. 
TopBases
13 bastg 20058 . . . . . . . 8  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
1412, 13ax-mp 5 . . . . . . 7  |-  ran  (,)  C_  ( topGen `  ran  (,) )
1514, 3sseqtr4i 3451 . . . . . 6  |-  ran  (,)  C_  K
1611, 15sstri 3427 . . . . 5  |-  ( (,) " ( { -oo }  X.  RR ) ) 
C_  K
17 retop 21860 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  Top
183, 17eqeltri 2545 . . . . . . 7  |-  K  e. 
Top
1918elexi 3041 . . . . . 6  |-  K  e. 
_V
2019elpw2 4565 . . . . 5  |-  ( ( (,) " ( { -oo }  X.  RR ) )  e.  ~P K 
<->  ( (,) " ( { -oo }  X.  RR ) )  C_  K
)
2116, 20mpbir 214 . . . 4  |-  ( (,) " ( { -oo }  X.  RR ) )  e.  ~P K
22 bndth.3 . . . . . 6  |-  ( ph  ->  J  e.  Comp )
23 rncmp 20488 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
2422, 1, 23syl2anc 673 . . . . 5  |-  ( ph  ->  ( Kt  ran  F )  e. 
Comp )
256cmpsub 20492 . . . . . 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 676 . . . . 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 215 . . . 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 4198 . . . . . . . 8  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  U. u  =  U. ( (,) " ( { -oo }  X.  RR ) ) )
2911unissi 4213 . . . . . . . . . 10  |-  U. ( (,) " ( { -oo }  X.  RR ) ) 
C_  U. ran  (,)
30 unirnioo 11759 . . . . . . . . . 10  |-  RR  =  U. ran  (,)
3129, 30sseqtr4i 3451 . . . . . . . . 9  |-  U. ( (,) " ( { -oo }  X.  RR ) ) 
C_  RR
32 id 22 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  RR )
33 ltp1 10465 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <  ( x  +  1 ) )
34 ressxr 9702 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
35 peano2re 9824 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
3634, 35sseldi 3416 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR* )
37 elioomnf 11754 . . . . . . . . . . . . 13  |-  ( ( x  +  1 )  e.  RR*  ->  ( x  e.  ( -oo (,) ( x  +  1
) )  <->  ( x  e.  RR  /\  x  < 
( x  +  1 ) ) ) )
3836, 37syl 17 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  ( -oo (,) ( x  +  1 ) )  <->  ( x  e.  RR  /\  x  < 
( x  +  1 ) ) ) )
3932, 33, 38mpbir2and 936 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  ( -oo (,) (
x  +  1 ) ) )
40 df-ov 6311 . . . . . . . . . . . 12  |-  ( -oo (,) ( x  +  1 ) )  =  ( (,) `  <. -oo , 
( x  +  1 ) >. )
41 mnfxr 11437 . . . . . . . . . . . . . . . 16  |- -oo  e.  RR*
4241elexi 3041 . . . . . . . . . . . . . . 15  |- -oo  e.  _V
4342snid 3988 . . . . . . . . . . . . . 14  |- -oo  e.  { -oo }
44 opelxpi 4871 . . . . . . . . . . . . . 14  |-  ( ( -oo  e.  { -oo }  /\  ( x  + 
1 )  e.  RR )  ->  <. -oo ,  ( x  +  1 ) >.  e.  ( { -oo }  X.  RR ) )
4543, 35, 44sylancr 676 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  <. -oo , 
( x  +  1 ) >.  e.  ( { -oo }  X.  RR ) )
46 ioof 11757 . . . . . . . . . . . . . . 15  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
47 ffun 5742 . . . . . . . . . . . . . . 15  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
4846, 47ax-mp 5 . . . . . . . . . . . . . 14  |-  Fun  (,)
49 snssi 4107 . . . . . . . . . . . . . . . . 17  |-  ( -oo  e.  RR*  ->  { -oo }  C_ 
RR* )
5041, 49ax-mp 5 . . . . . . . . . . . . . . . 16  |-  { -oo } 
C_  RR*
51 xpss12 4945 . . . . . . . . . . . . . . . 16  |-  ( ( { -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
5250, 34, 51mp2an 686 . . . . . . . . . . . . . . 15  |-  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* )
5346fdmi 5746 . . . . . . . . . . . . . . 15  |-  dom  (,)  =  ( RR*  X.  RR* )
5452, 53sseqtr4i 3451 . . . . . . . . . . . . . 14  |-  ( { -oo }  X.  RR )  C_  dom  (,)
55 funfvima2 6158 . . . . . . . . . . . . . 14  |-  ( ( Fun  (,)  /\  ( { -oo }  X.  RR )  C_  dom  (,) )  ->  ( <. -oo ,  ( x  +  1 )
>.  e.  ( { -oo }  X.  RR )  -> 
( (,) `  <. -oo ,  ( x  + 
1 ) >. )  e.  ( (,) " ( { -oo }  X.  RR ) ) ) )
5648, 54, 55mp2an 686 . . . . . . . . . . . . 13  |-  ( <. -oo ,  ( x  + 
1 ) >.  e.  ( { -oo }  X.  RR )  ->  ( (,) `  <. -oo ,  ( x  +  1 ) >.
)  e.  ( (,) " ( { -oo }  X.  RR ) ) )
5745, 56syl 17 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  ( (,) `  <. -oo ,  ( x  +  1 )
>. )  e.  ( (,) " ( { -oo }  X.  RR ) ) )
5840, 57syl5eqel 2553 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  ( -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( { -oo }  X.  RR ) ) )
59 elunii 4195 . . . . . . . . . . 11  |-  ( ( x  e.  ( -oo (,) ( x  +  1 ) )  /\  ( -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( { -oo }  X.  RR ) ) )  ->  x  e.  U. ( (,) " ( { -oo }  X.  RR ) ) )
6039, 58, 59syl2anc 673 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  U. ( (,) " ( { -oo }  X.  RR ) ) )
6160ssriv 3422 . . . . . . . . 9  |-  RR  C_  U. ( (,) " ( { -oo }  X.  RR ) )
6231, 61eqssi 3434 . . . . . . . 8  |-  U. ( (,) " ( { -oo }  X.  RR ) )  =  RR
6328, 62syl6eq 2521 . . . . . . 7  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  U. u  =  RR )
6463sseq2d 3446 . . . . . 6  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  ( ran  F  C_  U. u  <->  ran 
F  C_  RR )
)
65 pweq 3945 . . . . . . . 8  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  ~P u  =  ~P ( (,) " ( { -oo }  X.  RR ) ) )
6665ineq1d 3624 . . . . . . 7  |-  ( u  =  ( (,) " ( { -oo }  X.  RR ) )  ->  ( ~P u  i^i  Fin )  =  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )
6766rexeqdv 2980 . . . . . 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 327 . . . . 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 3132 . . . 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 64 . . 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 468 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )
73 elin 3608 . . . . . . 7  |-  ( v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  <->  ( v  e.  ~P ( (,) " ( { -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7472, 73sylib 201 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  (
v  e.  ~P ( (,) " ( { -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7574adantrr 731 . . . . 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 470 . . . 4  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  v  e.  Fin )
7774simpld 466 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ~P ( (,) " ( { -oo }  X.  RR ) ) )
7877elpwid 3952 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  C_  ( (,) " ( { -oo }  X.  RR ) ) )
7950sseli 3414 . . . . . . . . . . . 12  |-  ( u  e.  { -oo }  ->  u  e.  RR* )
8079adantr 472 . . . . . . . . . . 11  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  u  e.  RR* )
8134sseli 3414 . . . . . . . . . . . 12  |-  ( w  e.  RR  ->  w  e.  RR* )
8281adantl 473 . . . . . . . . . . 11  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  w  e.  RR* )
83 mnflt 11448 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  -> -oo  <  w )
84 xrltnle 9719 . . . . . . . . . . . . . . . 16  |-  ( ( -oo  e.  RR*  /\  w  e.  RR* )  ->  ( -oo  <  w  <->  -.  w  <_ -oo ) )
8541, 81, 84sylancr 676 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  ->  ( -oo  <  w  <->  -.  w  <_ -oo ) )
8683, 85mpbid 215 . . . . . . . . . . . . . 14  |-  ( w  e.  RR  ->  -.  w  <_ -oo )
8786adantl 473 . . . . . . . . . . . . 13  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  -.  w  <_ -oo )
88 elsni 3985 . . . . . . . . . . . . . . 15  |-  ( u  e.  { -oo }  ->  u  = -oo )
8988adantr 472 . . . . . . . . . . . . . 14  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  u  = -oo )
9089breq2d 4407 . . . . . . . . . . . . 13  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( w  <_  u 
<->  w  <_ -oo )
)
9187, 90mtbird 308 . . . . . . . . . . . 12  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  -.  w  <_  u )
92 ioo0 11686 . . . . . . . . . . . . . 14  |-  ( ( u  e.  RR*  /\  w  e.  RR* )  ->  (
( u (,) w
)  =  (/)  <->  w  <_  u ) )
9379, 81, 92syl2an 485 . . . . . . . . . . . . 13  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =  (/) 
<->  w  <_  u )
)
9493necon3abid 2679 . . . . . . . . . . . 12  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =/=  (/) 
<->  -.  w  <_  u
) )
9591, 94mpbird 240 . . . . . . . . . . 11  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  ( u (,) w )  =/=  (/) )
96 df-ioo 11664 . . . . . . . . . . . 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 11471 . . . . . . . . . . . 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 11471 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <_  x ) )
10196, 97, 98, 99, 100ixxub 11681 . . . . . . . . . . 11  |-  ( ( u  e.  RR*  /\  w  e.  RR*  /\  ( u (,) w )  =/=  (/) )  ->  sup (
( u (,) w
) ,  RR* ,  <  )  =  w )
10280, 82, 95, 101syl3anc 1292 . . . . . . . . . 10  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  =  w )
103 simpr 468 . . . . . . . . . 10  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  w  e.  RR )
104102, 103eqeltrd 2549 . . . . . . . . 9  |-  ( ( u  e.  { -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  e.  RR )
105104rgen2 2818 . . . . . . . 8  |-  A. u  e.  { -oo } A. w  e.  RR  sup ( ( u (,) w ) ,  RR* ,  <  )  e.  RR
106 fveq2 5879 . . . . . . . . . . . 12  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( (,) `  <. u ,  w >. ) )
107 df-ov 6311 . . . . . . . . . . . 12  |-  ( u (,) w )  =  ( (,) `  <. u ,  w >. )
108106, 107syl6eqr 2523 . . . . . . . . . . 11  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( u (,) w ) )
109108supeq1d 7978 . . . . . . . . . 10  |-  ( z  =  <. u ,  w >.  ->  sup ( ( (,) `  z ) ,  RR* ,  <  )  =  sup ( ( u (,) w ) ,  RR* ,  <  ) )
110109eleq1d 2533 . . . . . . . . 9  |-  ( z  =  <. u ,  w >.  ->  ( sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR  <->  sup (
( u (,) w
) ,  RR* ,  <  )  e.  RR ) )
111110ralxp 4981 . . . . . . . 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 214 . . . . . . 7  |-  A. z  e.  ( { -oo }  X.  RR ) sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR
113 ffn 5739 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
11446, 113ax-mp 5 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
115 supeq1 7977 . . . . . . . . . 10  |-  ( w  =  ( (,) `  z
)  ->  sup (
w ,  RR* ,  <  )  =  sup ( ( (,) `  z ) ,  RR* ,  <  )
)
116115eleq1d 2533 . . . . . . . . 9  |-  ( w  =  ( (,) `  z
)  ->  ( sup ( w ,  RR* ,  <  )  e.  RR  <->  sup ( ( (,) `  z
) ,  RR* ,  <  )  e.  RR ) )
117116ralima 6163 . . . . . . . 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 686 . . . . . . 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 214 . . . . . 6  |-  A. w  e.  ( (,) " ( { -oo }  X.  RR ) ) sup (
w ,  RR* ,  <  )  e.  RR
120 ssralv 3479 . . . . . 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 21 . . . . 5  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
122121adantrr 731 . . . 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 10575 . . . 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 673 . . 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 779 . . . . . . . 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 3418 . . . . . . 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 4194 . . . . . . . 8  |-  ( z  e.  U. v  <->  E. w  e.  v  z  e.  w )
128 r19.29r 2913 . . . . . . . . . 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 4360 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (,) " ( { -oo }  X.  RR ) )  C_  ~P RR 
<-> 
U. ( (,) " ( { -oo }  X.  RR ) )  C_  RR )
13031, 129mpbir 214 . . . . . . . . . . . . . . . . . 18  |-  ( (,) " ( { -oo }  X.  RR ) ) 
C_  ~P RR
131783ad2ant1 1051 . . . . . . . . . . . . . . . . . . 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 1057 . . . . . . . . . . . . . . . . . . 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 3419 . . . . . . . . . . . . . . . . . 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 3416 . . . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . . 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 1058 . . . . . . . . . . . . . . . 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 3419 . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  /\  w  e.  v )  ->  sup (
w ,  RR* ,  <  )  e.  RR )
139138adantrl 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v ) )  ->  sup ( w ,  RR* ,  <  )  e.  RR )
1401393adant3 1050 . . . . . . . . . . . . . . 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 1056 . . . . . . . . . . . . . . 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 3430 . . . . . . . . . . . . . . . 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 11635 . . . . . . . . . . . . . . . 16  |-  ( ( w  C_  RR*  /\  z  e.  w )  ->  z  <_  sup ( w , 
RR* ,  <  ) )
144142, 136, 143syl2anc 673 . . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . 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 9809 . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 660 . . . . . . . . . . . 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 2871 . . . . . . . . . . 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 735 . . . . . . . . . 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 444 . . . . . . . 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 483 . . . . . . 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 478 . . . . . 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 2812 . . . . 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 5739 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
1578, 156syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
158157ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( { -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  F  Fn  X )
159 breq1 4398 . . . . . . 7  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
160159ralrn 6040 . . . . . 6  |-  ( F  Fn  X  ->  ( A. z  e.  ran  F  z  <_  x  <->  A. y  e.  X  ( F `  y )  <_  x
) )
161158, 160syl 17 . . . . 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 222 . . . 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 2858 . . 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 2875 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395    X. cxp 4837   dom cdm 4839   ran crn 4840   "cima 4842   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556   1c1 9558    + caddc 9560   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694   (,)cioo 11660   ↾t crest 15397   topGenctg 15414   Topctop 19994  TopOnctopon 19995   TopBasesctb 19997    Cn ccn 20317   Compccmp 20478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-ioo 11664  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-cmp 20479
This theorem is referenced by:  evth  22065
  Copyright terms: Public domain W3C validator