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Theorem ivthALT 26339
Description: An alternate proof of the Intermediate Value Theorem ivth 19352 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 994 . . . . . 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 18924 . . . . . 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 5594 . . . . 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 981 . . 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 18862 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
873adant3 978 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
983ad2ant1 979 . . . . . . 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 964 . . . . . . . . . . . . . 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 554 . . . . . . . . . . . 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 1150 . . . . . . . . . . 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 981 . . . . . . . . . 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 454 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  Fun  F )
16 fdm 5596 . . . . . . . . . . . . 13  |-  ( F : D --> CC  ->  dom 
F  =  D )
1716sseq2d 3377 . . . . . . . . . . . 12  |-  ( F : D --> CC  ->  ( ( A [,] B
)  C_  dom  F  <->  ( A [,] B )  C_  D
) )
1817biimparc 475 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  ( A [,] B )  C_  dom  F )
1915, 18jca 520 . . . . . . . . . 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 5663 . . . . . . . . 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 18796 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  Top
24 simp332 1112 . . . . . . . . . 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 18797 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
2625restuni 17227 . . . . . . . . . 10  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( F
" ( A [,] B ) )  C_  RR )  ->  ( F
" ( A [,] B ) )  = 
U. ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) )
2723, 24, 26sylancr 646 . . . . . . . . 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 5652 . . . . . . . . 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 203 . . . . . . 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 1111 . . . . . . . . . . . 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 3368 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
33 eqid 2437 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
34 eqid 2437 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
3533cnfldtop 18819 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  Top
3633cnfldtopon 18818 . . . . . . . . . . . . . . . . . . . 20  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736toponunii 16998 . . . . . . . . . . . . . . . . . . 19  |-  CC  =  U. ( TopOpen ` fld )
3837restid 13662 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3935, 38ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4039eqcomi 2441 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
4133, 34, 40cncfcn 18940 . . . . . . . . . . . . . . 15  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
4232, 41mpan2 654 . . . . . . . . . . . . . 14  |-  ( D 
C_  CC  ->  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
43423ad2ant2 980 . . . . . . . . . . . . 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 981 . . . . . . . . . . . 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 2513 . . . . . . . . . . 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 994 . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 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 17226 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
4936, 47, 48sylancr 646 . . . . . . . . . . . . 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 16993 . . . . . . . . . . . . 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 3385 . . . . . . . . . . 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 2437 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  D )  =  U. ( ( TopOpen ` fld )t  D )
5453cnrest 17350 . . . . . . . . . . 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 644 . . . . . . . . . 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 9072 . . . . . . . . . . . . . 14  |-  CC  e.  _V
58 ssexg 4350 . . . . . . . . . . . . . 14  |-  ( ( D  C_  CC  /\  CC  e.  _V )  ->  D  e.  _V )
5947, 57, 58sylancl 645 . . . . . . . . . . . . 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 17230 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 10993 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
63623adant3 978 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  ( A [,] B )  C_  RR )
64633ad2ant1 979 . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
6633, 65rerest 18836 . . . . . . . . . . . . 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 2469 . . . . . . . . . . 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 6097 . . . . . . . . . 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 2513 . . . . . . . . 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 4892 . . . . . . . . . . . 12  |-  ( F
" ( A [,] B ) )  =  ran  ( F  |`  ( A [,] B ) )
7372eqimss2i 3404 . . . . . . . . . . 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 9048 . . . . . . . . . . 11  |-  RR  C_  CC
7624, 75syl6ss 3361 . . . . . . . . . 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 17351 . . . . . . . . . 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 1185 . . . . . . . . 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 203 . . . . . . . 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 18836 . . . . . . . . . 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 6098 . . . . . . . 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 2513 . . . . . . 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 2437 . . . . . . . 8  |-  U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )
8584cnconn 17486 . . . . . . 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 1185 . . . . . 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 18860 . . . . . . . . 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 980 . . . . . . . 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 981 . . . . . . 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 981 . . . . . 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 203 . . . . 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 988 . . . . . . . . 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 9135 . . . . . . . 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 989 . . . . . . . . 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 9135 . . . . . . . 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 9164 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
9796imp 420 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  A  <_  B )
98973adantl3 1116 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B )  ->  A  <_  B
)
99983adant3 978 . . . . . . . 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 11014 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
10193, 95, 99, 100syl3anc 1185 . . . . . . 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 5975 . . . . . . 7  |-  ( ( Fun  F  /\  ( A [,] B )  C_  dom  F )  ->  ( A  e.  ( A [,] B )  ->  ( F `  A )  e.  ( F " ( A [,] B ) ) ) )
10320, 101, 102sylc 59 . . . . . 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 11015 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
10593, 95, 99, 104syl3anc 1185 . . . . . . 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 5975 . . . . . . 7  |-  ( ( Fun  F  /\  ( A [,] B )  C_  dom  F )  ->  ( B  e.  ( A [,] B )  ->  ( F `  B )  e.  ( F " ( A [,] B ) ) ) )
10720, 105, 106sylc 59 . . . . . 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 6089 . . . . . . . 8  |-  ( x  =  ( F `  A )  ->  (
x [,] y )  =  ( ( F `
 A ) [,] y ) )
109108sseq1d 3376 . . . . . . 7  |-  ( x  =  ( F `  A )  ->  (
( x [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] y )  C_  ( F " ( A [,] B ) ) ) )
110 oveq2 6090 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  (
( F `  A
) [,] y )  =  ( ( F `
 A ) [,] ( F `  B
) ) )
111110sseq1d 3376 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A ) [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] ( F `  B
) )  C_  ( F " ( A [,] B ) ) ) )
112109, 111rspc2v 3059 . . . . . 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 644 . . . . 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 10997 . . . . . . . 8  |-  ( ( F `  A ) (,) ( F `  B ) )  C_  ( ( F `  A ) [,] ( F `  B )
)
116115sseli 3345 . . . . . . 7  |-  ( U  e.  ( ( F `
 A ) (,) ( F `  B
) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1171163ad2ant3 981 . . . . . 6  |-  ( ( F  e.  ( D
-cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1181173ad2ant3 981 . . . . 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 981 . . . 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 3350 . . 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 5779 . . 3  |-  ( ( Fun  F  /\  U  e.  ( F " ( A [,] B ) ) )  ->  E. x  e.  ( A [,] B
) ( F `  x )  =  U )
1226, 120, 121syl2anc 644 . 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 961 . . . . . . . 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 735 . . . . . . . . . . . 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 3350 . . . . . . . . . . . . . 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 1113 . . . . . . . . . . . . . . . 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 9135 . . . . . . . . . . . . . . . . 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 3350 . . . . . . . . . . . . . . . . . 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 9135 . . . . . . . . . . . . . . . . 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 10958 . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . 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 203 . . . . . . . . . . . . . . 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 971 . . . . . . . . . . . . . 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 9209 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 2620 . . . . . . . . . . 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 2618 . . . . . . . . . 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 5729 . . . . . . . . . 10  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
140138, 139nsyl 116 . . . . . . . . 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 990 . . . . . . . . . . . . . 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 972 . . . . . . . . . . . . . 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 9210 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 2620 . . . . . . . . . . 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 2618 . . . . . . . . . 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 5729 . . . . . . . . . 10  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
148146, 147nsyl 116 . . . . . . . . 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 1005 . . . . . . . . 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 26317 . . . . . . . 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 425 . . . . . . 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 521 . . . . . 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 941 . . . . . 6  |-  ( ( x  e.  RR*  /\  A  <  x  /\  x  < 
B )  <->  ( x  e.  RR*  /\  ( A  <  x  /\  x  <  B ) ) )
154152, 153syl6ibr 220 . . . . 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 9131 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
156 rexr 9131 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  RR* )
157 elicc3 26321 . . . . . . . . 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 465 . . . . . . . 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 978 . . . . . . 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 979 . . . . . 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 687 . . . . 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 10957 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
163155, 156, 162syl2an 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A (,) B )  <-> 
( x  e.  RR*  /\  A  <  x  /\  x  <  B ) ) )
1641633adant3 978 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
1651643ad2ant1 979 . . . . 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 261 . . . 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 449 . . . . 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 521 . . 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 2816 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   _Vcvv 2957    C_ wss 3321   U.cuni 4016   class class class wbr 4213   dom cdm 4879   ran crn 4880    |` cres 4881   "cima 4882   Fun wfun 5449   -->wf 5451   -onto->wfo 5453   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   RR*cxr 9120    < clt 9121    <_ cle 9122   (,)cioo 10917   [,]cicc 10920   ↾t crest 13649   TopOpenctopn 13650   topGenctg 13666  ℂfldccnfld 16704   Topctop 16959  TopOnctopon 16960    Cn ccn 17289   Conccon 17475   -cn->ccncf 18907
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-icc 10924  df-fz 11045  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-plusg 13543  df-mulr 13544  df-starv 13545  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-rest 13651  df-topn 13652  df-topgen 13668  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cld 17084  df-cn 17292  df-cnp 17293  df-con 17476  df-xms 18351  df-ms 18352  df-cncf 18909
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