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Theorem ivthALT 28530
Description: An alternate proof of the Intermediate Value Theorem ivth 20938 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 1024 . . . . . 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 20469 . . . . . 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 5561 . . . . 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 1011 . . 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 20407 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
873adant3 1008 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
983ad2ant1 1009 . . . . . . 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 994 . . . . . . . . . . . . . 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 569 . . . . . . . . . . . 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 1183 . . . . . . . . . . 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 1011 . . . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  Fun  F )
16 fdm 5563 . . . . . . . . . . . . 13  |-  ( F : D --> CC  ->  dom 
F  =  D )
1716sseq2d 3384 . . . . . . . . . . . 12  |-  ( F : D --> CC  ->  ( ( A [,] B
)  C_  dom  F  <->  ( A [,] B )  C_  D
) )
1817biimparc 487 . . . . . . . . . . 11  |-  ( ( ( A [,] B
)  C_  D  /\  F : D --> CC )  ->  ( A [,] B )  C_  dom  F )
1915, 18jca 532 . . . . . . . . . 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 5629 . . . . . . . . 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 20340 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  Top
24 simp332 1142 . . . . . . . . . 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 20341 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
2625restuni 18766 . . . . . . . . . 10  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( F
" ( A [,] B ) )  C_  RR )  ->  ( F
" ( A [,] B ) )  = 
U. ( ( topGen ` 
ran  (,) )t  ( F "
( A [,] B
) ) ) )
2723, 24, 26sylancr 663 . . . . . . . . 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 5618 . . . . . . . . 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 1141 . . . . . . . . . . . 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 3375 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
33 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
34 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
3533cnfldtop 20363 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  Top
3633cnfldtopon 20362 . . . . . . . . . . . . . . . . . . . 20  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736toponunii 18537 . . . . . . . . . . . . . . . . . . 19  |-  CC  =  U. ( TopOpen ` fld )
3837restid 14372 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4039eqcomi 2447 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
4133, 34, 40cncfcn 20485 . . . . . . . . . . . . . . 15  |-  ( ( D  C_  CC  /\  CC  C_  CC )  ->  ( D -cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
4232, 41mpan2 671 . . . . . . . . . . . . . 14  |-  ( D 
C_  CC  ->  ( D
-cn-> CC )  =  ( ( ( TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
43423ad2ant2 1010 . . . . . . . . . . . . 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 1011 . . . . . . . . . . . 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 2519 . . . . . . . . . . 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 1024 . . . . . . . . . . . 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 1025 . . . . . . . . . . . . . 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 18765 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  D  C_  CC )  ->  (
( TopOpen ` fld )t  D )  e.  (TopOn `  D ) )
4936, 47, 48sylancr 663 . . . . . . . . . . . . 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 18532 . . . . . . . . . . . . 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 3392 . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  D )  =  U. ( ( TopOpen ` fld )t  D )
5453cnrest 18889 . . . . . . . . . . 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 661 . . . . . . . . . 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 9363 . . . . . . . . . . . . . 14  |-  CC  e.  _V
58 ssexg 4438 . . . . . . . . . . . . . 14  |-  ( ( D  C_  CC  /\  CC  e.  _V )  ->  D  e.  _V )
5947, 57, 58sylancl 662 . . . . . . . . . . . . 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 18769 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 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 11377 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
63623adant3 1008 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  ( A [,] B )  C_  RR )
64633ad2ant1 1009 . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
6633, 65rerest 20381 . . . . . . . . . . . . 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 2475 . . . . . . . . . . 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 6106 . . . . . . . . . 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 2519 . . . . . . . . 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 4853 . . . . . . . . . . . 12  |-  ( F
" ( A [,] B ) )  =  ran  ( F  |`  ( A [,] B ) )
7372eqimss2i 3411 . . . . . . . . . . 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 9339 . . . . . . . . . . 11  |-  RR  C_  CC
7624, 75syl6ss 3368 . . . . . . . . . 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 18890 . . . . . . . . . 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 1218 . . . . . . . . 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 20381 . . . . . . . . . 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 6107 . . . . . . . 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 2519 . . . . . . 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 2443 . . . . . . . 8  |-  U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )
8584cnconn 19026 . . . . . . 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 1218 . . . . . 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 20405 . . . . . . . . 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 1010 . . . . . . . 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 1011 . . . . . . 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 1011 . . . . . 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 1018 . . . . . . . . 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 9433 . . . . . . . 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 1019 . . . . . . . . 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 9433 . . . . . . . 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 9463 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
9796imp 429 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  A  <_  B )
98973adantl3 1146 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B )  ->  A  <_  B
)
99983adant3 1008 . . . . . . . 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 11401 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
10193, 95, 99, 100syl3anc 1218 . . . . . . 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 5953 . . . . . . 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 11402 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
10593, 95, 99, 104syl3anc 1218 . . . . . . 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 5953 . . . . . . 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 6098 . . . . . . . 8  |-  ( x  =  ( F `  A )  ->  (
x [,] y )  =  ( ( F `
 A ) [,] y ) )
109108sseq1d 3383 . . . . . . 7  |-  ( x  =  ( F `  A )  ->  (
( x [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] y )  C_  ( F " ( A [,] B ) ) ) )
110 oveq2 6099 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  (
( F `  A
) [,] y )  =  ( ( F `
 A ) [,] ( F `  B
) ) )
111110sseq1d 3383 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A ) [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] ( F `  B
) )  C_  ( F " ( A [,] B ) ) ) )
112109, 111rspc2v 3079 . . . . . 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 661 . . . . 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 11381 . . . . . . . 8  |-  ( ( F `  A ) (,) ( F `  B ) )  C_  ( ( F `  A ) [,] ( F `  B )
)
116115sseli 3352 . . . . . . 7  |-  ( U  e.  ( ( F `
 A ) (,) ( F `  B
) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1171163ad2ant3 1011 . . . . . 6  |-  ( ( F  e.  ( D
-cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1181173ad2ant3 1011 . . . . 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 1011 . . . 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 3357 . . 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 5743 . . 3  |-  ( ( Fun  F  /\  U  e.  ( F " ( A [,] B ) ) )  ->  E. x  e.  ( A [,] B
) ( F `  x )  =  U )
1226, 120, 121syl2anc 661 . 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 991 . . . . . . . 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 756 . . . . . . . . . . . 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 3357 . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . . . 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 9433 . . . . . . . . . . . . . . . . 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 3357 . . . . . . . . . . . . . . . . . 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 9433 . . . . . . . . . . . . . . . . 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 11341 . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . 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 9509 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 2626 . . . . . . . . . . 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 2624 . . . . . . . . . 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 5691 . . . . . . . . . 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 1020 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  U  e.  RR )
142133simp3d 1002 . . . . . . . . . . . . . 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 9510 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 2626 . . . . . . . . . . 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 2624 . . . . . . . . . 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 5691 . . . . . . . . . 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 1035 . . . . . . . . 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 28508 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B ) 
C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F
" ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  /\  ( ( x  e. 
RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U ) )  ->  ( A  < 
x  /\  x  <  B ) )
151150ex 434 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( ( ( x  e.  RR*  /\  A  <_  B  /\  ( x  =  A  \/  ( A  <  x  /\  x  <  B )  \/  x  =  B ) )  /\  ( F `  x )  =  U )  -> 
( A  <  x  /\  x  <  B ) ) )
152124, 151jcad 533 . . . . . 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 969 . . . . . 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 9429 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
156 rexr 9429 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  RR* )
157 elicc3 28512 . . . . . . . . 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 477 . . . . . . . 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 1008 . . . . . . 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 1009 . . . . . 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 704 . . . . 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 11340 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
163155, 156, 162syl2an 477 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A (,) B )  <-> 
( x  e.  RR*  /\  A  <  x  /\  x  <  B ) ) )
1641633adant3 1008 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
1651643ad2ant1 1009 . . . . 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 461 . . . . 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 533 . . 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 2825 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( E. x  e.  ( A [,] B
) ( F `  x )  =  U  ->  E. x  e.  ( A (,) B ) ( F `  x
)  =  U ) )
171122, 170mpd 15 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  ->  E. x  e.  ( A (,) B ) ( F `  x )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   _Vcvv 2972    C_ wss 3328   U.cuni 4091   class class class wbr 4292   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   Fun wfun 5412   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   RR*cxr 9417    < clt 9418    <_ cle 9419   (,)cioo 11300   [,]cicc 11303   ↾t crest 14359   TopOpenctopn 14360   topGenctg 14376  ℂfldccnfld 17818   Topctop 18498  TopOnctopon 18499    Cn ccn 18828   Conccon 19015   -cn->ccncf 20452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-mulr 14252  df-starv 14253  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-rest 14361  df-topn 14362  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-cn 18831  df-cnp 18832  df-con 19016  df-xms 19895  df-ms 19896  df-cncf 20454
This theorem is referenced by: (None)
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