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Theorem ivthALT 30393
Description: An alternate proof of the Intermediate Value Theorem ivth 22032 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
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 1030 . . . . . 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 21563 . . . . . 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 5715 . . . . 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 1017 . . 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 21501 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
873adant3 1014 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
983ad2ant1 1015 . . . . . . 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 1000 . . . . . . . . . . . . . 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 567 . . . . . . . . . . . 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 1190 . . . . . . . . . . 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 1017 . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  Fun  F )
16 fdm 5717 . . . . . . . . . . . . 13  |-  ( F : D --> CC  ->  dom 
F  =  D )
1716sseq2d 3517 . . . . . . . . . . . 12  |-  ( F : D --> CC  ->  ( ( A [,] B
)  C_  dom  F  <->  ( A [,] B )  C_  D
) )
1817biimparc 485 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  ( A [,] B )  C_  dom  F )
1915, 18jca 530 . . . . . . . . . 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 5786 . . . . . . . . 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 21434 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  Top
24 simp332 1148 . . . . . . . . . 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 21435 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
2625restuni 19830 . . . . . . . . . 10  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( F
" ( A [,] B ) )  C_  RR )  ->  ( F
" ( A [,] B ) )  = 
U. ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) )
2723, 24, 26sylancr 661 . . . . . . . . 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 5775 . . . . . . . . 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 1147 . . . . . . . . . . . 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 3508 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
33 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
34 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
3533cnfldtop 21457 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  Top
3633cnfldtopon 21456 . . . . . . . . . . . . . . . . . . . 20  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736toponunii 19600 . . . . . . . . . . . . . . . . . . 19  |-  CC  =  U. ( TopOpen ` fld )
3837restid 14923 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4039eqcomi 2467 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
4133, 34, 40cncfcn 21579 . . . . . . . . . . . . . . 15  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
4232, 41mpan2 669 . . . . . . . . . . . . . 14  |-  ( D 
C_  CC  ->  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
43423ad2ant2 1016 . . . . . . . . . . . . 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 1017 . . . . . . . . . . . 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 2544 . . . . . . . . . . 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 1030 . . . . . . . . . . . 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 1031 . . . . . . . . . . . . . 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 19829 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
4936, 47, 48sylancr 661 . . . . . . . . . . . . 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 19595 . . . . . . . . . . . . 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 3525 . . . . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  D )  =  U. ( ( TopOpen ` fld )t  D )
5453cnrest 19953 . . . . . . . . . . 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 659 . . . . . . . . . 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 9562 . . . . . . . . . . . . . 14  |-  CC  e.  _V
58 ssexg 4583 . . . . . . . . . . . . . 14  |-  ( ( D  C_  CC  /\  CC  e.  _V )  ->  D  e.  _V )
5947, 57, 58sylancl 660 . . . . . . . . . . . . 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 19833 . . . . . . . . . . . . 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 1226 . . . . . . . . . . . 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 11609 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
63623adant3 1014 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  ( A [,] B )  C_  RR )
64633ad2ant1 1015 . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
6633, 65rerest 21475 . . . . . . . . . . . . 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 2495 . . . . . . . . . . 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 6285 . . . . . . . . . 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 2544 . . . . . . . . 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 5001 . . . . . . . . . . . 12  |-  ( F
" ( A [,] B ) )  =  ran  ( F  |`  ( A [,] B ) )
7372eqimss2i 3544 . . . . . . . . . . 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 9538 . . . . . . . . . . 11  |-  RR  C_  CC
7624, 75syl6ss 3501 . . . . . . . . . 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 19954 . . . . . . . . . 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 1226 . . . . . . . . 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 21475 . . . . . . . . . 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 6286 . . . . . . . 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 2544 . . . . . . 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 2454 . . . . . . . 8  |-  U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )
8584cnconn 20089 . . . . . . 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 1226 . . . . . 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 21499 . . . . . . . . 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 1016 . . . . . . . 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 1017 . . . . . . 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 1017 . . . . . 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 1024 . . . . . . . . 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 9632 . . . . . . . 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 1025 . . . . . . . . 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 9632 . . . . . . . 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 9662 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
9796imp 427 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  A  <_  B )
98973adantl3 1152 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B )  ->  A  <_  B
)
99983adant3 1014 . . . . . . . 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 11639 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
10193, 95, 99, 100syl3anc 1226 . . . . . . 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 6123 . . . . . . 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 11640 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
10593, 95, 99, 104syl3anc 1226 . . . . . . 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 6123 . . . . . . 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 6277 . . . . . . . 8  |-  ( x  =  ( F `  A )  ->  (
x [,] y )  =  ( ( F `
 A ) [,] y ) )
109108sseq1d 3516 . . . . . . 7  |-  ( x  =  ( F `  A )  ->  (
( x [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] y )  C_  ( F " ( A [,] B ) ) ) )
110 oveq2 6278 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  (
( F `  A
) [,] y )  =  ( ( F `
 A ) [,] ( F `  B
) ) )
111110sseq1d 3516 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A ) [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] ( F `  B
) )  C_  ( F " ( A [,] B ) ) ) )
112109, 111rspc2v 3216 . . . . . 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 659 . . . . 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 11613 . . . . . . . 8  |-  ( ( F `  A ) (,) ( F `  B ) )  C_  ( ( F `  A ) [,] ( F `  B )
)
116115sseli 3485 . . . . . . 7  |-  ( U  e.  ( ( F `
 A ) (,) ( F `  B
) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1171163ad2ant3 1017 . . . . . 6  |-  ( ( F  e.  ( D
-cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1181173ad2ant3 1017 . . . . 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 1017 . . . 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 3490 . . 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 5900 . . 3  |-  ( ( Fun  F  /\  U  e.  ( F " ( A [,] B ) ) )  ->  E. x  e.  ( A [,] B
) ( F `  x )  =  U )
1226, 120, 121syl2anc 659 . 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 997 . . . . . . . 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 755 . . . . . . . . . . . 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 3490 . . . . . . . . . . . . . 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 1149 . . . . . . . . . . . . . . . 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 9632 . . . . . . . . . . . . . . . . 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 3490 . . . . . . . . . . . . . . . . . 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 9632 . . . . . . . . . . . . . . . . 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 11573 . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . 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 1007 . . . . . . . . . . . . . 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 9709 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 2747 . . . . . . . . . . 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 2656 . . . . . . . . . 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 5848 . . . . . . . . . 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 1026 . . . . . . . . . . . . . 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 1008 . . . . . . . . . . . . . 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 9710 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 2747 . . . . . . . . . . 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 2656 . . . . . . . . . 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 5848 . . . . . . . . . 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 1041 . . . . . . . . 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 30371 . . . . . . . 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 432 . . . . . . 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 531 . . . . . 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 975 . . . . . 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 9628 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
156 rexr 9628 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  RR* )
157 elicc3 30375 . . . . . . . . 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 475 . . . . . . . 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 1014 . . . . . . 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 1015 . . . . . 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 702 . . . . 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 11572 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
163155, 156, 162syl2an 475 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A (,) B )  <-> 
( x  e.  RR*  /\  A  <  x  /\  x  <  B ) ) )
1641633adant3 1014 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
1651643ad2ant1 1015 . . . . 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 459 . . . . 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 531 . . 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 2925 . 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 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    C_ wss 3461   U.cuni 4235   class class class wbr 4439   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Fun wfun 5564   -->wf 5566   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   RR*cxr 9616    < clt 9617    <_ cle 9618   (,)cioo 11532   [,]cicc 11535   ↾t crest 14910   TopOpenctopn 14911   topGenctg 14927  ℂfldccnfld 18615   Topctop 19561  TopOnctopon 19562    Cn ccn 19892   Conccon 20078   -cn->ccncf 21546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-rest 14912  df-topn 14913  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-cn 19895  df-cnp 19896  df-con 20079  df-xms 20989  df-ms 20990  df-cncf 21548
This theorem is referenced by: (None)
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