Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ivthALT Structured version   Unicode version

Theorem ivthALT 28439
Description: An alternate proof of the Intermediate Value Theorem ivth 20838 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
ivthALT  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  E. x  e.  ( A (,) B ) ( F `  x )  =  U )
Distinct variable groups:    x, A    x, B    x, D    x, F    x, U

Proof of Theorem ivthALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp31 1019 . . . . . 6  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  F  e.  ( D -cn-> CC ) )
2 cncff 20369 . . . . . 6  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
31, 2syl 16 . . . . 5  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  F : D --> CC )
4 ffun 5558 . . . . 5  |-  ( F : D --> CC  ->  Fun 
F )
53, 4syl 16 . . . 4  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  Fun  F )
653ad2ant3 1006 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  Fun  F )
7 iccconn 20307 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
873adant3 1003 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
983ad2ant1 1004 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
10 simpr1 989 . . . . . . . . . . . . . 14  |-  ( ( D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  F  e.  ( D -cn-> CC ) )
1110, 2syl 16 . . . . . . . . . . . . 13  |-  ( ( D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  F : D --> CC )
1211anim2i 566 . . . . . . . . . . . 12  |-  ( ( ( A [,] B
)  C_  D  /\  ( D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( A [,] B )  C_  D  /\  F : D --> CC ) )
13123impb 1178 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  (
( A [,] B
)  C_  D  /\  F : D --> CC ) )
14133ad2ant3 1006 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( A [,] B )  C_  D  /\  F : D --> CC ) )
154adantl 463 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  Fun  F )
16 fdm 5560 . . . . . . . . . . . . 13  |-  ( F : D --> CC  ->  dom 
F  =  D )
1716sseq2d 3381 . . . . . . . . . . . 12  |-  ( F : D --> CC  ->  ( ( A [,] B
)  C_  dom  F  <->  ( A [,] B )  C_  D
) )
1817biimparc 484 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  ( A [,] B )  C_  dom  F )
1915, 18jca 529 . . . . . . . . . 10  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  ( Fun  F  /\  ( A [,] B
)  C_  dom  F ) )
2014, 19syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( Fun  F  /\  ( A [,] B ) 
C_  dom  F )
)
21 fores 5626 . . . . . . . . 9  |-  ( ( Fun  F  /\  ( A [,] B )  C_  dom  F )  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B
) -onto-> ( F "
( A [,] B
) ) )
2220, 21syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F  |`  ( A [,] B ) ) : ( A [,] B ) -onto-> ( F
" ( A [,] B ) ) )
23 retop 20240 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  Top
24 simp332 1137 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F " ( A [,] B ) ) 
C_  RR )
25 uniretop 20241 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
2625restuni 18666 . . . . . . . . . 10  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( F
" ( A [,] B ) )  C_  RR )  ->  ( F
" ( A [,] B ) )  = 
U. ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) )
2723, 24, 26sylancr 658 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F " ( A [,] B ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) ) )
28 foeq3 5615 . . . . . . . . 9  |-  ( ( F " ( A [,] B ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  ->  ( ( F  |`  ( A [,] B
) ) : ( A [,] B )
-onto-> ( F " ( A [,] B ) )  <-> 
( F  |`  ( A [,] B ) ) : ( A [,] B ) -onto-> U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) ) ) )
2927, 28syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( F  |`  ( A [,] B ) ) : ( A [,] B ) -onto-> ( F " ( A [,] B ) )  <-> 
( F  |`  ( A [,] B ) ) : ( A [,] B ) -onto-> U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) ) ) )
3022, 29mpbid 210 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F  |`  ( A [,] B ) ) : ( A [,] B ) -onto-> U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) ) )
31 simp331 1136 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  F  e.  ( D -cn->
CC ) )
32 ssid 3372 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
33 eqid 2441 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
34 eqid 2441 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
3533cnfldtop 20263 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  Top
3633cnfldtopon 20262 . . . . . . . . . . . . . . . . . . . 20  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736toponunii 18437 . . . . . . . . . . . . . . . . . . 19  |-  CC  =  U. ( TopOpen ` fld )
3837restid 14368 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4039eqcomi 2445 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
4133, 34, 40cncfcn 20385 . . . . . . . . . . . . . . 15  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
4232, 41mpan2 666 . . . . . . . . . . . . . 14  |-  ( D 
C_  CC  ->  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
43423ad2ant2 1005 . . . . . . . . . . . . 13  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
44433ad2ant3 1006 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( D -cn-> CC )  =  ( ( (
TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
4531, 44eleqtrd 2517 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  F  e.  ( (
( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
46 simp31 1019 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( A [,] B
)  C_  D )
47 simp32 1020 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  D  C_  CC )
48 resttopon 18665 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
4936, 47, 48sylancr 658 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
50 toponuni 18432 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )t  D )  e.  (TopOn `  D )  ->  D  =  U. ( ( TopOpen ` fld )t  D
) )
5149, 50syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  D  =  U. (
( TopOpen ` fld )t  D ) )
5246, 51sseqtrd 3389 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( A [,] B
)  C_  U. (
( TopOpen ` fld )t  D ) )
53 eqid 2441 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  D )  =  U. ( ( TopOpen ` fld )t  D )
5453cnrest 18789 . . . . . . . . . . 11  |-  ( ( F  e.  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) )  /\  ( A [,] B )  C_  U. ( ( TopOpen ` fld )t  D ) )  -> 
( F  |`  ( A [,] B ) )  e.  ( ( ( ( TopOpen ` fld )t  D )t  ( A [,] B ) )  Cn  ( TopOpen ` fld ) ) )
5545, 52, 54syl2anc 656 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F  |`  ( A [,] B ) )  e.  ( ( ( ( TopOpen ` fld )t  D )t  ( A [,] B ) )  Cn  ( TopOpen ` fld ) ) )
5635a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( TopOpen ` fld )  e.  Top )
57 cnex 9359 . . . . . . . . . . . . . 14  |-  CC  e.  _V
58 ssexg 4435 . . . . . . . . . . . . . 14  |-  ( ( D  C_  CC  /\  CC  e.  _V )  ->  D  e.  _V )
5947, 57, 58sylancl 657 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  D  e.  _V )
60 restabs 18669 . . . . . . . . . . . . 13  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( A [,] B
)  C_  D  /\  D  e.  _V )  ->  ( ( ( TopOpen ` fld )t  D
)t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
6156, 46, 59, 60syl3anc 1213 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( TopOpen ` fld )t  D
)t  ( A [,] B
) )  =  ( ( TopOpen ` fld )t  ( A [,] B ) ) )
62 iccssre 11373 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
63623adant3 1003 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  ( A [,] B )  C_  RR )
64633ad2ant1 1004 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( A [,] B
)  C_  RR )
65 eqid 2441 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
6633, 65rerest 20281 . . . . . . . . . . . . 13  |-  ( ( A [,] B ) 
C_  RR  ->  ( (
TopOpen ` fld )t  ( A [,] B
) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
6764, 66syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( TopOpen ` fld )t  ( A [,] B ) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B
) ) )
6861, 67eqtrd 2473 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( TopOpen ` fld )t  D
)t  ( A [,] B
) )  =  ( ( topGen `  ran  (,) )t  ( A [,] B ) ) )
6968oveq1d 6105 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( (
TopOpen ` fld )t  D )t  ( A [,] B ) )  Cn  ( TopOpen ` fld ) )  =  ( ( ( topGen `  ran  (,) )t  ( A [,] B
) )  Cn  ( TopOpen
` fld
) ) )
7055, 69eleqtrd 2517 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F  |`  ( A [,] B ) )  e.  ( ( (
topGen `  ran  (,) )t  ( A [,] B ) )  Cn  ( TopOpen ` fld ) ) )
7136a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( TopOpen ` fld )  e.  (TopOn `  CC ) )
72 df-ima 4849 . . . . . . . . . . . 12  |-  ( F
" ( A [,] B ) )  =  ran  ( F  |`  ( A [,] B ) )
7372eqimss2i 3408 . . . . . . . . . . 11  |-  ran  ( F  |`  ( A [,] B ) )  C_  ( F " ( A [,] B ) )
7473a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  ran  ( F  |`  ( A [,] B ) ) 
C_  ( F "
( A [,] B
) ) )
75 ax-resscn 9335 . . . . . . . . . . 11  |-  RR  C_  CC
7624, 75syl6ss 3365 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F " ( A [,] B ) ) 
C_  CC )
77 cnrest2 18790 . . . . . . . . . 10  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( F  |`  ( A [,] B ) ) 
C_  ( F "
( A [,] B
) )  /\  ( F " ( A [,] B ) )  C_  CC )  ->  ( ( F  |`  ( A [,] B ) )  e.  ( ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  Cn  ( TopOpen ` fld ) )  <->  ( F  |`  ( A [,] B
) )  e.  ( ( ( topGen `  ran  (,) )t  ( A [,] B
) )  Cn  (
( TopOpen ` fld )t  ( F "
( A [,] B
) ) ) ) ) )
7871, 74, 76, 77syl3anc 1213 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( F  |`  ( A [,] B ) )  e.  ( ( ( topGen `  ran  (,) )t  ( A [,] B ) )  Cn  ( TopOpen ` fld ) )  <->  ( F  |`  ( A [,] B
) )  e.  ( ( ( topGen `  ran  (,) )t  ( A [,] B
) )  Cn  (
( TopOpen ` fld )t  ( F "
( A [,] B
) ) ) ) ) )
7970, 78mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F  |`  ( A [,] B ) )  e.  ( ( (
topGen `  ran  (,) )t  ( A [,] B ) )  Cn  ( ( TopOpen ` fld )t  ( F " ( A [,] B ) ) ) ) )
8033, 65rerest 20281 . . . . . . . . . 10  |-  ( ( F " ( A [,] B ) ) 
C_  RR  ->  ( (
TopOpen ` fld )t  ( F " ( A [,] B ) ) )  =  ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) ) )
8124, 80syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( TopOpen ` fld )t  ( F "
( A [,] B
) ) )  =  ( ( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) ) )
8281oveq2d 6106 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( topGen ` 
ran  (,) )t  ( A [,] B ) )  Cn  ( ( TopOpen ` fld )t  ( F "
( A [,] B
) ) ) )  =  ( ( (
topGen `  ran  (,) )t  ( A [,] B ) )  Cn  ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) ) )
8379, 82eleqtrd 2517 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F  |`  ( A [,] B ) )  e.  ( ( (
topGen `  ran  (,) )t  ( A [,] B ) )  Cn  ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) ) )
84 eqid 2441 . . . . . . . 8  |-  U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )
8584cnconn 18926 . . . . . . 7  |-  ( ( ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con  /\  ( F  |`  ( A [,] B ) ) : ( A [,] B ) -onto-> U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  /\  ( F  |`  ( A [,] B ) )  e.  ( ( ( topGen `  ran  (,) )t  ( A [,] B ) )  Cn  ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) ) )  ->  ( ( topGen `
 ran  (,) )t  ( F " ( A [,] B ) ) )  e.  Con )
869, 30, 83, 85syl3anc 1213 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  e.  Con )
87 reconn 20305 . . . . . . . . 9  |-  ( ( F " ( A [,] B ) ) 
C_  RR  ->  ( ( ( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  e.  Con  <->  A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B ) ) ( x [,] y
)  C_  ( F " ( A [,] B
) ) ) )
88873ad2ant2 1005 . . . . . . . 8  |-  ( ( F  e.  ( D
-cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) )  ->  ( (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  e.  Con  <->  A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B ) ) ( x [,] y
)  C_  ( F " ( A [,] B
) ) ) )
89883ad2ant3 1006 . . . . . . 7  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  (
( ( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  e.  Con  <->  A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B ) ) ( x [,] y
)  C_  ( F " ( A [,] B
) ) ) )
90893ad2ant3 1006 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) )  e. 
Con 
<-> 
A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B ) ) ( x [,] y
)  C_  ( F " ( A [,] B
) ) ) )
9186, 90mpbid 210 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B
) ) ( x [,] y )  C_  ( F " ( A [,] B ) ) )
92 simp11 1013 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  A  e.  RR )
9392rexrd 9429 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  A  e.  RR* )
94 simp12 1014 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  B  e.  RR )
9594rexrd 9429 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  B  e.  RR* )
96 ltle 9459 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
9796imp 429 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  A  <_  B )
98973adantl3 1141 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B )  ->  A  <_  B
)
99983adant3 1003 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  A  <_  B )
100 lbicc2 11397 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
10193, 95, 99, 100syl3anc 1213 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  A  e.  ( A [,] B ) )
102 funfvima2 5950 . . . . . . 7  |-  ( ( Fun  F  /\  ( A [,] B )  C_  dom  F )  ->  ( A  e.  ( A [,] B )  ->  ( F `  A )  e.  ( F " ( A [,] B ) ) ) )
10320, 101, 102sylc 60 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  A
)  e.  ( F
" ( A [,] B ) ) )
104 ubicc2 11398 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
10593, 95, 99, 104syl3anc 1213 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  B  e.  ( A [,] B ) )
106 funfvima2 5950 . . . . . . 7  |-  ( ( Fun  F  /\  ( A [,] B )  C_  dom  F )  ->  ( B  e.  ( A [,] B )  ->  ( F `  B )  e.  ( F " ( A [,] B ) ) ) )
10720, 105, 106sylc 60 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  B
)  e.  ( F
" ( A [,] B ) ) )
108 oveq1 6097 . . . . . . . 8  |-  ( x  =  ( F `  A )  ->  (
x [,] y )  =  ( ( F `
 A ) [,] y ) )
109108sseq1d 3380 . . . . . . 7  |-  ( x  =  ( F `  A )  ->  (
( x [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] y )  C_  ( F " ( A [,] B ) ) ) )
110 oveq2 6098 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  (
( F `  A
) [,] y )  =  ( ( F `
 A ) [,] ( F `  B
) ) )
111110sseq1d 3380 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A ) [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] ( F `  B
) )  C_  ( F " ( A [,] B ) ) ) )
112109, 111rspc2v 3076 . . . . . 6  |-  ( ( ( F `  A
)  e.  ( F
" ( A [,] B ) )  /\  ( F `  B )  e.  ( F "
( A [,] B
) ) )  -> 
( A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B ) ) ( x [,] y
)  C_  ( F " ( A [,] B
) )  ->  (
( F `  A
) [,] ( F `
 B ) ) 
C_  ( F "
( A [,] B
) ) ) )
113103, 107, 112syl2anc 656 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( A. x  e.  ( F " ( A [,] B ) ) A. y  e.  ( F " ( A [,] B ) ) ( x [,] y
)  C_  ( F " ( A [,] B
) )  ->  (
( F `  A
) [,] ( F `
 B ) ) 
C_  ( F "
( A [,] B
) ) ) )
11491, 113mpd 15 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( F `  A ) [,] ( F `  B )
)  C_  ( F " ( A [,] B
) ) )
115 ioossicc 11377 . . . . . . . 8  |-  ( ( F `  A ) (,) ( F `  B ) )  C_  ( ( F `  A ) [,] ( F `  B )
)
116115sseli 3349 . . . . . . 7  |-  ( U  e.  ( ( F `
 A ) (,) ( F `  B
) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1171163ad2ant3 1006 . . . . . 6  |-  ( ( F  e.  ( D
-cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1181173ad2ant3 1006 . . . . 5  |-  ( ( ( A [,] B
)  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1191183ad2ant3 1006 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B
) ) )
120114, 119sseldd 3354 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  e.  ( F " ( A [,] B
) ) )
121 fvelima 5740 . . 3  |-  ( ( Fun  F  /\  U  e.  ( F " ( A [,] B ) ) )  ->  E. x  e.  ( A [,] B
) ( F `  x )  =  U )
1226, 120, 121syl2anc 656 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  E. x  e.  ( A [,] B ) ( F `  x )  =  U )
123 simpl1 986 . . . . . . . 8  |-  ( ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U )  ->  x  e.  RR* )
124123a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U )  ->  x  e.  RR* ) )
125 simprr 751 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  ( F `  x )  =  U )
12624, 103sseldd 3354 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  A
)  e.  RR )
127 simp333 1138 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  e.  ( ( F `  A ) (,) ( F `  B
) ) )
128126rexrd 9429 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  A
)  e.  RR* )
12924, 107sseldd 3354 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  B
)  e.  RR )
130129rexrd 9429 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  B
)  e.  RR* )
131 elioo2 11337 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  A
)  e.  RR*  /\  ( F `  B )  e.  RR* )  ->  ( U  e.  ( ( F `  A ) (,) ( F `  B
) )  <->  ( U  e.  RR  /\  ( F `
 A )  < 
U  /\  U  <  ( F `  B ) ) ) )
132128, 130, 131syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( U  e.  ( ( F `  A
) (,) ( F `
 B ) )  <-> 
( U  e.  RR  /\  ( F `  A
)  <  U  /\  U  <  ( F `  B ) ) ) )
133127, 132mpbid 210 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( U  e.  RR  /\  ( F `  A
)  <  U  /\  U  <  ( F `  B ) ) )
134133simp2d 996 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( F `  A
)  <  U )
135126, 134gtned 9505 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  =/=  ( F `  A ) )
136135adantr 462 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  U  =/=  ( F `  A )
)
137125, 136eqnetrd 2624 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  ( F `  x )  =/=  ( F `  A )
)
138137neneqd 2622 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  -.  ( F `  x )  =  ( F `  A ) )
139 fveq2 5688 . . . . . . . . . 10  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
140138, 139nsyl 121 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  -.  x  =  A )
141 simp13 1015 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  e.  RR )
142133simp3d 997 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  <  ( F `  B ) )
143141, 142ltned 9506 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  =/=  ( F `  B ) )
144143adantr 462 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  U  =/=  ( F `  B )
)
145125, 144eqnetrd 2624 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  ( F `  x )  =/=  ( F `  B )
)
146145neneqd 2622 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  -.  ( F `  x )  =  ( F `  B ) )
147 fveq2 5688 . . . . . . . . . 10  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
148146, 147nsyl 121 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  -.  x  =  B )
149 simprl3 1030 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )
150140, 148, 149ecase13d 28417 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  ( A  < 
x  /\  x  <  B ) )
151150ex 434 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U )  -> 
( A  <  x  /\  x  <  B ) ) )
152124, 151jcad 530 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U )  -> 
( x  e.  RR*  /\  ( A  <  x  /\  x  <  B ) ) ) )
153 3anass 964 . . . . . 6  |-  ( ( x  e.  RR*  /\  A  <  x  /\  x  < 
B )  <->  ( x  e.  RR*  /\  ( A  <  x  /\  x  <  B ) ) )
154152, 153syl6ibr 227 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U )  -> 
( x  e.  RR*  /\  A  <  x  /\  x  <  B ) ) )
155 rexr 9425 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
156 rexr 9425 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  RR* )
157 elicc3 28421 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,] B )  <->  ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) ) ) )
158155, 156, 157syl2an 474 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) ) ) )
1591583adant3 1003 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
x  e.  ( A [,] B )  <->  ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) ) ) )
1601593ad2ant1 1004 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( x  e.  ( A [,] B )  <-> 
( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) ) ) )
161160anbi1d 699 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( x  e.  ( A [,] B
)  /\  ( F `  x )  =  U )  <->  ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) ) )
162 elioo1 11336 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
163155, 156, 162syl2an 474 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A (,) B )  <-> 
( x  e.  RR*  /\  A  <  x  /\  x  <  B ) ) )
1641633adant3 1003 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
1651643ad2ant1 1004 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( x  e.  ( A (,) B )  <-> 
( x  e.  RR*  /\  A  <  x  /\  x  <  B ) ) )
166154, 161, 1653imtr4d 268 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( x  e.  ( A [,] B
)  /\  ( F `  x )  =  U )  ->  x  e.  ( A (,) B ) ) )
167 simpr 458 . . . . 5  |-  ( ( x  e.  ( A [,] B )  /\  ( F `  x )  =  U )  -> 
( F `  x
)  =  U )
168167a1i 11 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( x  e.  ( A [,] B
)  /\  ( F `  x )  =  U )  ->  ( F `  x )  =  U ) )
169166, 168jcad 530 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( x  e.  ( A [,] B
)  /\  ( F `  x )  =  U )  ->  ( x  e.  ( A (,) B
)  /\  ( F `  x )  =  U ) ) )
170169reximdv2 2823 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( E. x  e.  ( A [,] B
) ( F `  x )  =  U  ->  E. x  e.  ( A (,) B ) ( F `  x
)  =  U ) )
171122, 170mpd 15 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  E. x  e.  ( A (,) B ) ( F `  x )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 959    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    C_ wss 3325   U.cuni 4088   class class class wbr 4289   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   RR*cxr 9413    < clt 9414    <_ cle 9415   (,)cioo 11296   [,]cicc 11299   ↾t crest 14355   TopOpenctopn 14356   topGenctg 14372  ℂfldccnfld 17718   Topctop 18398  TopOnctopon 18399    Cn ccn 18728   Conccon 18915   -cn->ccncf 20352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-rest 14357  df-topn 14358  df-topgen 14378  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-cn 18731  df-cnp 18732  df-con 18916  df-xms 19795  df-ms 19796  df-cncf 20354
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator