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Theorem bndth 18985
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 18799 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
53, 4eqeltri 2508 . . . . . . 7  |-  K  e.  (TopOn `  RR )
65toponunii 16999 . . . . . 6  |-  RR  =  U. K
72, 6cnf 17312 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> RR )
81, 7syl 16 . . . 4  |-  ( ph  ->  F : X --> RR )
9 frn 5599 . . . 4  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . 3  |-  ( ph  ->  ran  F  C_  RR )
11 imassrn 5218 . . . . . 6  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  ran  (,)
12 retopbas 18796 . . . . . . . 8  |-  ran  (,)  e. 
TopBases
13 bastg 17033 . . . . . . . 8  |-  ( ran 
(,)  e.  TopBases  ->  ran  (,)  C_  ( topGen `  ran  (,) )
)
1412, 13ax-mp 8 . . . . . . 7  |-  ran  (,)  C_  ( topGen `  ran  (,) )
1514, 3sseqtr4i 3383 . . . . . 6  |-  ran  (,)  C_  K
1611, 15sstri 3359 . . . . 5  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  K
17 retop 18797 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  Top
183, 17eqeltri 2508 . . . . . . 7  |-  K  e. 
Top
1918elexi 2967 . . . . . 6  |-  K  e. 
_V
2019elpw2 4366 . . . . 5  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  e.  ~P K 
<->  ( (,) " ( {  -oo }  X.  RR ) )  C_  K
)
2116, 20mpbir 202 . . . 4  |-  ( (,) " ( {  -oo }  X.  RR ) )  e.  ~P K
22 bndth.3 . . . . . 6  |-  ( ph  ->  J  e.  Comp )
23 rncmp 17461 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K
) )  ->  ( Kt  ran  F )  e.  Comp )
2422, 1, 23syl2anc 644 . . . . 5  |-  ( ph  ->  ( Kt  ran  F )  e. 
Comp )
256cmpsub 17465 . . . . . 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 646 . . . . 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 203 . . . 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 4026 . . . . . . . 8  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  U. u  =  U. ( (,) " ( {  -oo }  X.  RR ) ) )
2911unissi 4040 . . . . . . . . . 10  |-  U. ( (,) " ( {  -oo }  X.  RR ) ) 
C_  U. ran  (,)
30 unirnioo 11006 . . . . . . . . . 10  |-  RR  =  U. ran  (,)
3129, 30sseqtr4i 3383 . . . . . . . . 9  |-  U. ( (,) " ( {  -oo }  X.  RR ) ) 
C_  RR
32 id 21 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  RR )
33 ltp1 9850 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <  ( x  +  1 ) )
34 ressxr 9131 . . . . . . . . . . . . . 14  |-  RR  C_  RR*
35 peano2re 9241 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
3634, 35sseldi 3348 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR* )
37 elioomnf 11001 . . . . . . . . . . . . 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 890 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  (  -oo (,) (
x  +  1 ) ) )
40 df-ov 6086 . . . . . . . . . . . 12  |-  (  -oo (,) ( x  +  1 ) )  =  ( (,) `  <.  -oo , 
( x  +  1 ) >. )
41 mnfxr 10716 . . . . . . . . . . . . . . . 16  |-  -oo  e.  RR*
4241elexi 2967 . . . . . . . . . . . . . . 15  |-  -oo  e.  _V
4342snid 3843 . . . . . . . . . . . . . 14  |-  -oo  e.  { 
-oo }
44 opelxpi 4912 . . . . . . . . . . . . . 14  |-  ( ( 
-oo  e.  {  -oo }  /\  ( x  +  1 )  e.  RR )  ->  <.  -oo ,  ( x  +  1 ) >.  e.  ( {  -oo }  X.  RR ) )
4543, 35, 44sylancr 646 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  <.  -oo , 
( x  +  1 ) >.  e.  ( {  -oo }  X.  RR ) )
46 ioof 11004 . . . . . . . . . . . . . . 15  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
47 ffun 5595 . . . . . . . . . . . . . . 15  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
4846, 47ax-mp 8 . . . . . . . . . . . . . 14  |-  Fun  (,)
49 snssi 3944 . . . . . . . . . . . . . . . . 17  |-  (  -oo  e.  RR*  ->  {  -oo }  C_ 
RR* )
5041, 49ax-mp 8 . . . . . . . . . . . . . . . 16  |-  {  -oo } 
C_  RR*
51 xpss12 4983 . . . . . . . . . . . . . . . 16  |-  ( ( {  -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( {  -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
5250, 34, 51mp2an 655 . . . . . . . . . . . . . . 15  |-  ( { 
-oo }  X.  RR )  C_  ( RR*  X.  RR* )
5346fdmi 5598 . . . . . . . . . . . . . . 15  |-  dom  (,)  =  ( RR*  X.  RR* )
5452, 53sseqtr4i 3383 . . . . . . . . . . . . . 14  |-  ( { 
-oo }  X.  RR )  C_  dom  (,)
55 funfvima2 5976 . . . . . . . . . . . . . 14  |-  ( ( Fun  (,)  /\  ( {  -oo }  X.  RR )  C_  dom  (,) )  ->  ( <.  -oo ,  ( x  +  1 )
>.  e.  ( {  -oo }  X.  RR )  -> 
( (,) `  <.  -oo
,  ( x  + 
1 ) >. )  e.  ( (,) " ( {  -oo }  X.  RR ) ) ) )
5648, 54, 55mp2an 655 . . . . . . . . . . . . 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 2522 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  (  -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )
59 elunii 4022 . . . . . . . . . . 11  |-  ( ( x  e.  (  -oo (,) ( x  +  1 ) )  /\  (  -oo (,) ( x  + 
1 ) )  e.  ( (,) " ( {  -oo }  X.  RR ) ) )  ->  x  e.  U. ( (,) " ( {  -oo }  X.  RR ) ) )
6039, 58, 59syl2anc 644 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  U. ( (,) " ( {  -oo }  X.  RR ) ) )
6160ssriv 3354 . . . . . . . . 9  |-  RR  C_  U. ( (,) " ( {  -oo }  X.  RR ) )
6231, 61eqssi 3366 . . . . . . . 8  |-  U. ( (,) " ( {  -oo }  X.  RR ) )  =  RR
6328, 62syl6eq 2486 . . . . . . 7  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  U. u  =  RR )
6463sseq2d 3378 . . . . . 6  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( ran  F  C_  U. u  <->  ran 
F  C_  RR )
)
65 pweq 3804 . . . . . . . 8  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ~P u  =  ~P ( (,) " ( {  -oo }  X.  RR ) ) )
6665ineq1d 3543 . . . . . . 7  |-  ( u  =  ( (,) " ( {  -oo }  X.  RR ) )  ->  ( ~P u  i^i  Fin )  =  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )
6766rexeqdv 2913 . . . . . 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 313 . . . . 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 3050 . . . 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 62 . . 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 449 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )
73 elin 3532 . . . . . . 7  |-  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  <->  ( v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7472, 73sylib 190 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  (
v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) )  /\  v  e.  Fin ) )
7574adantrr 699 . . . . 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 451 . . . 4  |-  ( (
ph  /\  ( v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v
) )  ->  v  e.  Fin )
7774simpld 447 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  e.  ~P ( (,) " ( {  -oo }  X.  RR ) ) )
7877elpwid 3810 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  v  C_  ( (,) " ( {  -oo }  X.  RR ) ) )
7950sseli 3346 . . . . . . . . . . . 12  |-  ( u  e.  {  -oo }  ->  u  e.  RR* )
8079adantr 453 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  u  e.  RR* )
8134sseli 3346 . . . . . . . . . . . 12  |-  ( w  e.  RR  ->  w  e.  RR* )
8281adantl 454 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  w  e.  RR* )
83 mnflt 10724 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  ->  -oo  <  w )
84 xrltnle 9146 . . . . . . . . . . . . . . . 16  |-  ( ( 
-oo  e.  RR*  /\  w  e.  RR* )  ->  (  -oo  <  w  <->  -.  w  <_  -oo ) )
8541, 81, 84sylancr 646 . . . . . . . . . . . . . . 15  |-  ( w  e.  RR  ->  (  -oo  <  w  <->  -.  w  <_  -oo ) )
8683, 85mpbid 203 . . . . . . . . . . . . . 14  |-  ( w  e.  RR  ->  -.  w  <_  -oo )
8786adantl 454 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  -.  w  <_  -oo )
88 elsni 3840 . . . . . . . . . . . . . . 15  |-  ( u  e.  {  -oo }  ->  u  =  -oo )
8988adantr 453 . . . . . . . . . . . . . 14  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  u  =  -oo )
9089breq2d 4226 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( w  <_  u 
<->  w  <_  -oo ) )
9187, 90mtbird 294 . . . . . . . . . . . 12  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  -.  w  <_  u )
92 ioo0 10943 . . . . . . . . . . . . . 14  |-  ( ( u  e.  RR*  /\  w  e.  RR* )  ->  (
( u (,) w
)  =  (/)  <->  w  <_  u ) )
9379, 81, 92syl2an 465 . . . . . . . . . . . . 13  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =  (/) 
<->  w  <_  u )
)
9493necon3abid 2636 . . . . . . . . . . . 12  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( ( u (,) w )  =/=  (/) 
<->  -.  w  <_  u
) )
9591, 94mpbird 225 . . . . . . . . . . 11  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  ( u (,) w )  =/=  (/) )
96 df-ioo 10922 . . . . . . . . . . . 12  |-  (,)  =  ( y  e.  RR* ,  z  e.  RR*  |->  { v  e.  RR*  |  (
y  <  v  /\  v  <  z ) } )
97 idd 23 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <  w ) )
98 xrltle 10744 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  w  e.  RR* )  ->  (
x  <  w  ->  x  <_  w ) )
99 idd 23 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <  x ) )
100 xrltle 10744 . . . . . . . . . . . 12  |-  ( ( u  e.  RR*  /\  x  e.  RR* )  ->  (
u  <  x  ->  u  <_  x ) )
10196, 97, 98, 99, 100ixxub 10939 . . . . . . . . . . 11  |-  ( ( u  e.  RR*  /\  w  e.  RR*  /\  ( u (,) w )  =/=  (/) )  ->  sup (
( u (,) w
) ,  RR* ,  <  )  =  w )
10280, 82, 95, 101syl3anc 1185 . . . . . . . . . 10  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  =  w )
103 simpr 449 . . . . . . . . . 10  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  w  e.  RR )
104102, 103eqeltrd 2512 . . . . . . . . 9  |-  ( ( u  e.  {  -oo }  /\  w  e.  RR )  ->  sup ( ( u (,) w ) , 
RR* ,  <  )  e.  RR )
105104rgen2 2804 . . . . . . . 8  |-  A. u  e.  {  -oo } A. w  e.  RR  sup ( ( u (,) w ) ,  RR* ,  <  )  e.  RR
106 fveq2 5730 . . . . . . . . . . . 12  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( (,) `  <. u ,  w >. ) )
107 df-ov 6086 . . . . . . . . . . . 12  |-  ( u (,) w )  =  ( (,) `  <. u ,  w >. )
108106, 107syl6eqr 2488 . . . . . . . . . . 11  |-  ( z  =  <. u ,  w >.  ->  ( (,) `  z
)  =  ( u (,) w ) )
109108supeq1d 7453 . . . . . . . . . 10  |-  ( z  =  <. u ,  w >.  ->  sup ( ( (,) `  z ) ,  RR* ,  <  )  =  sup ( ( u (,) w ) ,  RR* ,  <  ) )
110109eleq1d 2504 . . . . . . . . 9  |-  ( z  =  <. u ,  w >.  ->  ( sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR  <->  sup (
( u (,) w
) ,  RR* ,  <  )  e.  RR ) )
111110ralxp 5018 . . . . . . . 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 202 . . . . . . 7  |-  A. z  e.  ( {  -oo }  X.  RR ) sup (
( (,) `  z
) ,  RR* ,  <  )  e.  RR
113 ffn 5593 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
11446, 113ax-mp 8 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
115 supeq1 7452 . . . . . . . . . 10  |-  ( w  =  ( (,) `  z
)  ->  sup (
w ,  RR* ,  <  )  =  sup ( ( (,) `  z ) ,  RR* ,  <  )
)
116115eleq1d 2504 . . . . . . . . 9  |-  ( w  =  ( (,) `  z
)  ->  ( sup ( w ,  RR* ,  <  )  e.  RR  <->  sup ( ( (,) `  z
) ,  RR* ,  <  )  e.  RR ) )
117116ralima 5980 . . . . . . . 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 655 . . . . . . 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 202 . . . . . 6  |-  A. w  e.  ( (,) " ( {  -oo }  X.  RR ) ) sup (
w ,  RR* ,  <  )  e.  RR
120 ssralv 3409 . . . . . 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 1386 . . . . 5  |-  ( (
ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  ->  A. w  e.  v  sup (
w ,  RR* ,  <  )  e.  RR )
122121adantrr 699 . . . 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 9959 . . . 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 644 . . 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 739 . . . . . . . 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 3350 . . . . . . 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 4021 . . . . . . . 8  |-  ( z  e.  U. v  <->  E. w  e.  v  z  e.  w )
128 r19.29r 2849 . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (,) " ( { 
-oo }  X.  RR ) )  C_  ~P RR 
<-> 
U. ( (,) " ( {  -oo }  X.  RR ) )  C_  RR )
13031, 129mpbir 202 . . . . . . . . . . . . . . . . . 18  |-  ( (,) " ( {  -oo }  X.  RR ) ) 
C_  ~P RR
131783ad2ant1 979 . . . . . . . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . . . . . 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 3348 . . . . . . . . . . . . . . . . 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 3810 . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  w  e.  v )  ->  sup (
w ,  RR* ,  <  )  e.  RR )
139138adantrl 698 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin ) )  /\  ( x  e.  RR  /\  w  e.  v ) )  ->  sup ( w ,  RR* ,  <  )  e.  RR )
1401393adant3 978 . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . 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 3362 . . . . . . . . . . . . . . . 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 10905 . . . . . . . . . . . . . . . 16  |-  ( ( w  C_  RR*  /\  z  e.  w )  ->  z  <_  sup ( w , 
RR* ,  <  ) )
144142, 136, 143syl2anc 644 . . . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . . 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 9229 . . . . . . . . . . . . . 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 1156 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 2832 . . . . . . . . . . 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 703 . . . . . . . . . 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 31 . . . . . . . . 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 428 . . . . . . . 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 463 . . . . . . 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 458 . . . . . 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 2798 . . . . 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 5593 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
1578, 156syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
158157ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( ~P ( (,) " ( {  -oo }  X.  RR ) )  i^i  Fin )  /\  ran  F  C_  U. v ) )  /\  x  e.  RR )  ->  F  Fn  X )
159 breq1 4217 . . . . . . 7  |-  ( z  =  ( F `  y )  ->  (
z  <_  x  <->  ( F `  y )  <_  x
) )
160159ralrn 5875 . . . . . 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 207 . . . 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 2820 . . 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 2836 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   <.cop 3819   U.cuni 4017   class class class wbr 4214    X. cxp 4878   dom cdm 4880   ran crn 4881   "cima 4883   Fun wfun 5450    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Fincfn 7111   supcsup 7447   RRcr 8991   1c1 8993    + caddc 8995    -oocmnf 9120   RR*cxr 9121    < clt 9122    <_ cle 9123   (,)cioo 10918   ↾t crest 13650   topGenctg 13667   Topctop 16960  TopOnctopon 16961   TopBasesctb 16964    Cn ccn 17290   Compccmp 17451
This theorem is referenced by:  evth  18986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-ioo 10922  df-rest 13652  df-topgen 13669  df-top 16965  df-bases 16967  df-topon 16968  df-cn 17293  df-cmp 17452
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