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Theorem bndth 18936
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 18750 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
53, 4eqeltri 2474 . . . . . . 7  |-  K  e.  (TopOn `  RR )
65toponunii 16952 . . . . . 6  |-  RR  =  U. K
72, 6cnf 17264 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
81, 7syl 16 . . . 4  |-  ( ph  ->  F : X --> RR )
9 frn 5556 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . 3  |-  ( ph  ->  ran  F  C_  RR )
11 imassrn 5175 . . . . . 6  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  ran  (,)
12 retopbas 18747 . . . . . . . 8  |-  ran  (,)  e. 
TopBases
13 bastg 16986 . . . . . . . 8  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
1412, 13ax-mp 8 . . . . . . 7  |-  ran  (,)  C_  ( topGen `  ran  (,) )
1514, 3sseqtr4i 3341 . . . . . 6  |-  ran  (,)  C_  K
1611, 15sstri 3317 . . . . 5  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  K
17 retop 18748 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  Top
183, 17eqeltri 2474 . . . . . . 7  |-  K  e. 
Top
1918elexi 2925 . . . . . 6  |-  K  e. 
_V
2019elpw2 4324 . . . . 5  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  e.  ~P K 
<->  ( (,) " ( {  -oo }  X.  RR ) )  C_  K
)
2116, 20mpbir 201 . . . 4  |-  ( (,) " ( {  -oo }  X.  RR ) )  e.  ~P K
22 bndth.3 . . . . . 6  |-  ( ph  ->  J  e.  Comp )
23 rncmp 17413 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
2422, 1, 23syl2anc 643 . . . . 5  |-  ( ph  ->  ( Kt  ran  F )  e. 
Comp )
256cmpsub 17417 . . . . . 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 645 . . . . 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 202 . . . 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 3984 . . . . . . . 8  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  U. u  =  U. ( (,) " ( {  -oo }  X.  RR ) ) )
2911unissi 3998 . . . . . . . . . 10  |-  U. ( (,) " ( {  -oo }  X.  RR ) ) 
C_  U. ran  (,)
30 unirnioo 10960 . . . . . . . . . 10  |-  RR  =  U. ran  (,)
3129, 30sseqtr4i 3341 . . . . . . . . 9  |-  U. ( (,) " ( {  -oo }  X.  RR ) ) 
C_  RR
32 id 20 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  RR )
33 ltp1 9804 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <  ( x  +  1 ) )
34 ressxr 9085 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
35 peano2re 9195 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
3634, 35sseldi 3306 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR* )
37 elioomnf 10955 . . . . . . . . . . . . 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 889 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  (  -oo (,) (
x  +  1 ) ) )
40 df-ov 6043 . . . . . . . . . . . 12  |-  (  -oo (,) ( x  +  1 ) )  =  ( (,) `  <.  -oo , 
( x  +  1 ) >. )
41 mnfxr 10670 . . . . . . . . . . . . . . . 16  |-  -oo  e.  RR*
4241elexi 2925 . . . . . . . . . . . . . . 15  |-  -oo  e.  _V
4342snid 3801 . . . . . . . . . . . . . 14  |-  -oo  e.  { 
-oo }
44 opelxpi 4869 . . . . . . . . . . . . . 14  |-  ( ( 
-oo  e.  {  -oo }  /\  ( x  +  1 )  e.  RR )  ->  <.  -oo ,  ( x  +  1 ) >.  e.  ( {  -oo }  X.  RR ) )
4543, 35, 44sylancr 645 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  <.  -oo , 
( x  +  1 ) >.  e.  ( {  -oo }  X.  RR ) )
46 ioof 10958 . . . . . . . . . . . . . . 15  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
47 ffun 5552 . . . . . . . . . . . . . . 15  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
4846, 47ax-mp 8 . . . . . . . . . . . . . 14  |-  Fun  (,)
49 snssi 3902 . . . . . . . . . . . . . . . . 17  |-  (  -oo  e.  RR*  ->  {  -oo }  C_ 
RR* )
5041, 49ax-mp 8 . . . . . . . . . . . . . . . 16  |-  {  -oo } 
C_  RR*
51 xpss12 4940 . . . . . . . . . . . . . . . 16  |-  ( ( {  -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( {  -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
5250, 34, 51mp2an 654 . . . . . . . . . . . . . . 15  |-  ( { 
-oo }  X.  RR )  C_  ( RR*  X.  RR* )
5346fdmi 5555 . . . . . . . . . . . . . . 15  |-  dom  (,)  =  ( RR*  X.  RR* )
5452, 53sseqtr4i 3341 . . . . . . . . . . . . . 14  |-  ( { 
-oo }  X.  RR )  C_  dom  (,)
55 funfvima2 5933 . . . . . . . . . . . . . 14  |-  ( ( Fun  (,)  /\  ( {  -oo }  X.  RR )  C_  dom  (,) )  ->  ( <.  -oo ,  ( x  +  1 )
>.  e.  ( {  -oo }  X.  RR )  -> 
( (,) `  <.  -oo
,  ( x  + 
1 ) >. )  e.  ( (,) " ( {  -oo }  X.  RR ) ) ) )
5648, 54, 55mp2an 654 . . . . . . . . . . . . 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 2488 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  (  -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )
59 elunii 3980 . . . . . . . . . . 11  |-  ( ( x  e.  (  -oo (,) ( x  +  1 ) )  /\  (  -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )  ->  x  e.  U. ( (,) " ( {  -oo }  X.  RR ) ) )
6039, 58, 59syl2anc 643 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  U. ( (,) " ( {  -oo }  X.  RR ) ) )
6160ssriv 3312 . . . . . . . . 9  |-  RR  C_  U. ( (,) " ( {  -oo }  X.  RR ) )
6231, 61eqssi 3324 . . . . . . . 8  |-  U. ( (,) " ( {  -oo }  X.  RR ) )  =  RR
6328, 62syl6eq 2452 . . . . . . 7  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  U. u  =  RR )
6463sseq2d 3336 . . . . . 6  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( ran  F  C_  U. u  <->  ran 
F  C_  RR )
)
65 pweq 3762 . . . . . . . 8  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ~P u  =  ~P ( (,) " ( {  -oo }  X.  RR ) ) )
6665ineq1d 3501 . . . . . . 7  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( ~P u  i^i  Fin )  =  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )
6766rexeqdv 2871 . . . . . 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 312 . . . . 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 3008 . . . 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 61 . . 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 448 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )
73 elin 3490 . . . . . . 7  |-  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  <->  ( v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7472, 73sylib 189 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  (
v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7574adantrr 698 . . . . 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 450 . . . 4  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  v  e.  Fin )
7774simpld 446 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) ) )
7877elpwid 3768 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  C_  ( (,) " ( {  -oo }  X.  RR ) ) )
7950sseli 3304 . . . . . . . . . . . 12  |-  ( u  e.  {  -oo }  ->  u  e.  RR* )
8079adantr 452 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  u  e.  RR* )
8134sseli 3304 . . . . . . . . . . . 12  |-  ( w  e.  RR  ->  w  e.  RR* )
8281adantl 453 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  w  e.  RR* )
83 mnflt 10678 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  ->  -oo  <  w )
84 xrltnle 9100 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  w  e.  RR* )  ->  (  -oo  <  w  <->  -.  w  <_  -oo ) )
8541, 81, 84sylancr 645 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  ->  (  -oo  <  w  <->  -.  w  <_  -oo ) )
8683, 85mpbid 202 . . . . . . . . . . . . . 14  |-  ( w  e.  RR  ->  -.  w  <_  -oo )
8786adantl 453 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  -.  w  <_  -oo )
88 elsni 3798 . . . . . . . . . . . . . . 15  |-  ( u  e.  {  -oo }  ->  u  =  -oo )
8988adantr 452 . . . . . . . . . . . . . 14  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  u  =  -oo )
9089breq2d 4184 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( w  <_  u 
<->  w  <_  -oo ) )
9187, 90mtbird 293 . . . . . . . . . . . 12  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  -.  w  <_  u )
92 ioo0 10897 . . . . . . . . . . . . . 14  |-  ( ( u  e.  RR*  /\  w  e.  RR* )  ->  (
( u (,) w
)  =  (/)  <->  w  <_  u ) )
9379, 81, 92syl2an 464 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =  (/) 
<->  w  <_  u )
)
9493necon3abid 2600 . . . . . . . . . . . 12  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =/=  (/) 
<->  -.  w  <_  u
) )
9591, 94mpbird 224 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( u (,) w )  =/=  (/) )
96 df-ioo 10876 . . . . . . . . . . . 12  |-  (,)  =  ( y  e.  RR* ,  z  e.  RR*  |->  { v  e.  RR*  |  (
y  <  v  /\  v  <  z ) } )
97 idd 22 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <  w ) )
98 xrltle 10698 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <_  w ) )
99 idd 22 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <  x ) )
100 xrltle 10698 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <_  x ) )
10196, 97, 98, 99, 100ixxub 10893 . . . . . . . . . . 11  |-  ( ( u  e.  RR*  /\  w  e.  RR*  /\  ( u (,) w )  =/=  (/) )  ->  sup (
( u (,) w
) ,  RR* ,  <  )  =  w )
10280, 82, 95, 101syl3anc 1184 . . . . . . . . . 10  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  =  w )
103 simpr 448 . . . . . . . . . 10  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  w  e.  RR )
104102, 103eqeltrd 2478 . . . . . . . . 9  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  e.  RR )
105104rgen2 2762 . . . . . . . 8  |-  A. u  e.  {  -oo } A. w  e.  RR  sup ( ( u (,) w ) ,  RR* ,  <  )  e.  RR
106 fveq2 5687 . . . . . . . . . . . 12  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( (,) `  <. u ,  w >. ) )
107 df-ov 6043 . . . . . . . . . . . 12  |-  ( u (,) w )  =  ( (,) `  <. u ,  w >. )
108106, 107syl6eqr 2454 . . . . . . . . . . 11  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( u (,) w ) )
109108supeq1d 7409 . . . . . . . . . 10  |-  ( z  =  <. u ,  w >.  ->  sup ( ( (,) `  z ) ,  RR* ,  <  )  =  sup ( ( u (,) w ) ,  RR* ,  <  ) )
110109eleq1d 2470 . . . . . . . . 9  |-  ( z  =  <. u ,  w >.  ->  ( sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR  <->  sup (
( u (,) w
) ,  RR* ,  <  )  e.  RR ) )
111110ralxp 4975 . . . . . . . 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 201 . . . . . . 7  |-  A. z  e.  ( {  -oo }  X.  RR ) sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR
113 ffn 5550 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
11446, 113ax-mp 8 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
115 supeq1 7408 . . . . . . . . . 10  |-  ( w  =  ( (,) `  z
)  ->  sup (
w ,  RR* ,  <  )  =  sup ( ( (,) `  z ) ,  RR* ,  <  )
)
116115eleq1d 2470 . . . . . . . . 9  |-  ( w  =  ( (,) `  z
)  ->  ( sup ( w ,  RR* ,  <  )  e.  RR  <->  sup ( ( (,) `  z
) ,  RR* ,  <  )  e.  RR ) )
117116ralima 5937 . . . . . . . 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 654 . . . . . . 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 201 . . . . . 6  |-  A. w  e.  ( (,) " ( {  -oo }  X.  RR ) ) sup (
w ,  RR* ,  <  )  e.  RR
120 ssralv 3367 . . . . . 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, 120ee10 1382 . . . . 5  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
122121adantrr 698 . . . 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 9913 . . . 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 643 . . 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 738 . . . . . . . 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 3308 . . . . . . 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 3979 . . . . . . . 8  |-  ( z  e.  U. v  <->  E. w  e.  v  z  e.  w )
128 r19.29r 2807 . . . . . . . . . 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 4136 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  C_  ~P RR 
<-> 
U. ( (,) " ( {  -oo }  X.  RR ) )  C_  RR )
13031, 129mpbir 201 . . . . . . . . . . . . . . . . . 18  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  ~P RR
131783ad2ant1 978 . . . . . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . . . . . 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 3306 . . . . . . . . . . . . . . . . 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 3768 . . . . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  w  e.  v )  ->  sup (
w ,  RR* ,  <  )  e.  RR )
139138adantrl 697 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v ) )  ->  sup ( w ,  RR* ,  <  )  e.  RR )
1401393adant3 977 . . . . . . . . . . . . . . 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 983 . . . . . . . . . . . . . . 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 3320 . . . . . . . . . . . . . . . 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 10859 . . . . . . . . . . . . . . . 16  |-  ( ( w  C_  RR*  /\  z  e.  w )  ->  z  <_  sup ( w , 
RR* ,  <  ) )
144142, 136, 143syl2anc 643 . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . 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 9183 . . . . . . . . . . . . . 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 1155 . . . . . . . . . . . . 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 630 . . . . . . . . . . . 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 2790 . . . . . . . . . . 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 702 . . . . . . . . . 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 30 . . . . . . . . 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 427 . . . . . . . 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 462 . . . . . . 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 457 . . . . . 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 2756 . . . . 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 5550 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
1578, 156syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
158157ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  F  Fn  X )
159 breq1 4175 . . . . . . 7  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
160159ralrn 5832 . . . . . 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 206 . . . 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 2778 . . 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 2794 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   <.cop 3777   U.cuni 3975   class class class wbr 4172    X. cxp 4835   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   supcsup 7403   RRcr 8945   1c1 8947    + caddc 8949    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   (,)cioo 10872   ↾t crest 13603   topGenctg 13620   Topctop 16913  TopOnctopon 16914   TopBasesctb 16917    Cn ccn 17242   Compccmp 17403
This theorem is referenced by:  evth  18937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-ioo 10876  df-rest 13605  df-topgen 13622  df-top 16918  df-bases 16920  df-topon 16921  df-cn 17245  df-cmp 17404
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