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Theorem ivthALT 29758
Description: An alternate proof of the Intermediate Value Theorem ivth 21629 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 1032 . . . . . 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 21160 . . . . . 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 5733 . . . . 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 1019 . . 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 21098 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B
) )  e.  Con )
873adant3 1016 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
983ad2ant1 1017 . . . . . . 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 1002 . . . . . . . . . . . . . 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 1192 . . . . . . . . . . 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 1019 . . . . . . . . . 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 5735 . . . . . . . . . . . . 13  |-  ( F : D --> CC  ->  dom 
F  =  D )
1716sseq2d 3532 . . . . . . . . . . . 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 5804 . . . . . . . . 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 21031 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  e.  Top
24 simp332 1150 . . . . . . . . . 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 21032 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
2625restuni 19457 . . . . . . . . . 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 5793 . . . . . . . . 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 1149 . . . . . . . . . . . 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 3523 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
33 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
34 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  D )  =  ( ( TopOpen ` fld )t  D )
3533cnfldtop 21054 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  Top
3633cnfldtopon 21053 . . . . . . . . . . . . . . . . . . . 20  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736toponunii 19228 . . . . . . . . . . . . . . . . . . 19  |-  CC  =  U. ( TopOpen ` fld )
3837restid 14689 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
3935, 38ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4039eqcomi 2480 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
4133, 34, 40cncfcn 21176 . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . 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 1019 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) ) ) )  -> 
( D -cn-> CC )  =  ( ( (
TopOpen ` fld )t  D )  Cn  ( TopOpen
` fld
) ) )
4531, 44eleqtrd 2557 . . . . . . . . . . 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 1032 . . . . . . . . . . . 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 1033 . . . . . . . . . . . . . 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 19456 . . . . . . . . . . . . . 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 19223 . . . . . . . . . . . . 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 3540 . . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  U. (
( TopOpen ` fld )t  D )  =  U. ( ( TopOpen ` fld )t  D )
5453cnrest 19580 . . . . . . . . . . 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 9573 . . . . . . . . . . . . . 14  |-  CC  e.  _V
58 ssexg 4593 . . . . . . . . . . . . . 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 19460 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 11606 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
63623adant3 1016 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  ( A [,] B )  C_  RR )
64633ad2ant1 1017 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
6633, 65rerest 21072 . . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 6299 . . . . . . . . . 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 2557 . . . . . . . . 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 5012 . . . . . . . . . . . 12  |-  ( F
" ( A [,] B ) )  =  ran  ( F  |`  ( A [,] B ) )
7372eqimss2i 3559 . . . . . . . . . . 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 9549 . . . . . . . . . . 11  |-  RR  C_  CC
7624, 75syl6ss 3516 . . . . . . . . . 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 19581 . . . . . . . . . 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 1228 . . . . . . . . 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 21072 . . . . . . . . . 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 6300 . . . . . . . 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 2557 . . . . . . 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 2467 . . . . . . . 8  |-  U. (
( topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )  =  U. ( (
topGen `  ran  (,) )t  ( F " ( A [,] B ) ) )
8584cnconn 19717 . . . . . . 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 1228 . . . . . 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 21096 . . . . . . . . 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 1018 . . . . . . . 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 1019 . . . . . . 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 1019 . . . . . 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 1026 . . . . . . . . 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 9643 . . . . . . . 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 1027 . . . . . . . . 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 9643 . . . . . . . 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 9673 . . . . . . . . . . 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 1154 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B )  ->  A  <_  B
)
99983adant3 1016 . . . . . . . 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 11636 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
10193, 95, 99, 100syl3anc 1228 . . . . . . 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 6136 . . . . . . 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 11637 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
10593, 95, 99, 104syl3anc 1228 . . . . . . 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 6136 . . . . . . 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 6291 . . . . . . . 8  |-  ( x  =  ( F `  A )  ->  (
x [,] y )  =  ( ( F `
 A ) [,] y ) )
109108sseq1d 3531 . . . . . . 7  |-  ( x  =  ( F `  A )  ->  (
( x [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] y )  C_  ( F " ( A [,] B ) ) ) )
110 oveq2 6292 . . . . . . . 8  |-  ( y  =  ( F `  B )  ->  (
( F `  A
) [,] y )  =  ( ( F `
 A ) [,] ( F `  B
) ) )
111110sseq1d 3531 . . . . . . 7  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A ) [,] y
)  C_  ( F " ( A [,] B
) )  <->  ( ( F `  A ) [,] ( F `  B
) )  C_  ( F " ( A [,] B ) ) ) )
112109, 111rspc2v 3223 . . . . . 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 11610 . . . . . . . 8  |-  ( ( F `  A ) (,) ( F `  B ) )  C_  ( ( F `  A ) [,] ( F `  B )
)
116115sseli 3500 . . . . . . 7  |-  ( U  e.  ( ( F `
 A ) (,) ( F `  B
) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1171163ad2ant3 1019 . . . . . 6  |-  ( ( F  e.  ( D
-cn-> CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A
) (,) ( F `
 B ) ) )  ->  U  e.  ( ( F `  A ) [,] ( F `  B )
) )
1181173ad2ant3 1019 . . . . 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 1019 . . . 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 3505 . . 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 5919 . . 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 999 . . . . . . . 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 3505 . . . . . . . . . . . . . 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 1151 . . . . . . . . . . . . . . . 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 9643 . . . . . . . . . . . . . . . . 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 3505 . . . . . . . . . . . . . . . . . 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 9643 . . . . . . . . . . . . . . . . 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 11570 . . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 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 9719 . . . . . . . . . . . . 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 2760 . . . . . . . . . . 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 2669 . . . . . . . . . 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 5866 . . . . . . . . . 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 1028 . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . 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 9720 . . . . . . . . . . . . 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 2760 . . . . . . . . . . 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 2669 . . . . . . . . . 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 5866 . . . . . . . . . 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 1043 . . . . . . . . 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 29736 . . . . . . . 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 977 . . . . . 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 9639 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
156 rexr 9639 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  RR* )
157 elicc3 29740 . . . . . . . . 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 1016 . . . . . . 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 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 ) ) ) ) )  -> 
( 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 11569 . . . . . . . 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 1016 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR*  /\  A  < 
x  /\  x  <  B ) ) )
1651643ad2ant1 1017 . . . . 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 2934 . 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 972    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   U.cuni 4245   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5582   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   RR*cxr 9627    < clt 9628    <_ cle 9629   (,)cioo 11529   [,]cicc 11532   ↾t crest 14676   TopOpenctopn 14677   topGenctg 14693  ℂfldccnfld 18219   Topctop 19189  TopOnctopon 19190    Cn ccn 19519   Conccon 19706   -cn->ccncf 21143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-rest 14678  df-topn 14679  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-cn 19522  df-cnp 19523  df-con 19707  df-xms 20586  df-ms 20587  df-cncf 21145
This theorem is referenced by: (None)
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