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Theorem brofs 24628
Description: Binary relationship form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Assertion
Ref Expression
brofs  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
<. F ,  H >. ) ) ) )

Proof of Theorem brofs
Dummy variables  a 
b  c  d  e  f  g  h  p  q  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3796 . . . . 5  |-  ( a  =  A  ->  <. a ,  c >.  =  <. A ,  c >. )
21breq2d 4035 . . . 4  |-  ( a  =  A  ->  (
b  Btwn  <. a ,  c >.  <->  b  Btwn  <. A , 
c >. ) )
32anbi1d 685 . . 3  |-  ( a  =  A  ->  (
( b  Btwn  <. a ,  c >.  /\  f  Btwn  <. e ,  g
>. )  <->  ( b  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. ) ) )
4 opeq1 3796 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
54breq1d 4033 . . . 4  |-  ( a  =  A  ->  ( <. a ,  b >.Cgr <. e ,  f >.  <->  <. A ,  b >.Cgr <.
e ,  f >.
) )
65anbi1d 685 . . 3  |-  ( a  =  A  ->  (
( <. a ,  b
>.Cgr <. e ,  f
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  <->  ( <. A , 
b >.Cgr <. e ,  f
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. ) ) )
7 opeq1 3796 . . . . 5  |-  ( a  =  A  ->  <. a ,  d >.  =  <. A ,  d >. )
87breq1d 4033 . . . 4  |-  ( a  =  A  ->  ( <. a ,  d >.Cgr <. e ,  h >.  <->  <. A ,  d >.Cgr <. e ,  h >. ) )
98anbi1d 685 . . 3  |-  ( a  =  A  ->  (
( <. a ,  d
>.Cgr <. e ,  h >.  /\  <. b ,  d
>.Cgr <. f ,  h >. )  <->  ( <. A , 
d >.Cgr <. e ,  h >.  /\  <. b ,  d
>.Cgr <. f ,  h >. ) ) )
103, 6, 93anbi123d 1252 . 2  |-  ( a  =  A  ->  (
( ( b  Btwn  <.
a ,  c >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. a ,  b
>.Cgr <. e ,  f
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. a ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) )  <->  ( ( b 
Btwn  <. A ,  c
>.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  b
>.Cgr <. e ,  f
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) ) ) )
11 breq1 4026 . . . 4  |-  ( b  =  B  ->  (
b  Btwn  <. A , 
c >. 
<->  B  Btwn  <. A , 
c >. ) )
1211anbi1d 685 . . 3  |-  ( b  =  B  ->  (
( b  Btwn  <. A , 
c >.  /\  f  Btwn  <.
e ,  g >.
)  <->  ( B  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. ) ) )
13 opeq2 3797 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
1413breq1d 4033 . . . 4  |-  ( b  =  B  ->  ( <. A ,  b >.Cgr <. e ,  f >.  <->  <. A ,  B >.Cgr <.
e ,  f >.
) )
15 opeq1 3796 . . . . 5  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1615breq1d 4033 . . . 4  |-  ( b  =  B  ->  ( <. b ,  c >.Cgr <. f ,  g >.  <->  <. B ,  c >.Cgr <.
f ,  g >.
) )
1714, 16anbi12d 691 . . 3  |-  ( b  =  B  ->  (
( <. A ,  b
>.Cgr <. e ,  f
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  <->  ( <. A ,  B >.Cgr <. e ,  f
>.  /\  <. B ,  c
>.Cgr <. f ,  g
>. ) ) )
18 opeq1 3796 . . . . 5  |-  ( b  =  B  ->  <. b ,  d >.  =  <. B ,  d >. )
1918breq1d 4033 . . . 4  |-  ( b  =  B  ->  ( <. b ,  d >.Cgr <. f ,  h >.  <->  <. B ,  d >.Cgr <. f ,  h >. ) )
2019anbi2d 684 . . 3  |-  ( b  =  B  ->  (
( <. A ,  d
>.Cgr <. e ,  h >.  /\  <. b ,  d
>.Cgr <. f ,  h >. )  <->  ( <. A , 
d >.Cgr <. e ,  h >.  /\  <. B ,  d
>.Cgr <. f ,  h >. ) ) )
2112, 17, 203anbi123d 1252 . 2  |-  ( b  =  B  ->  (
( ( b  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  b >.Cgr <. e ,  f >.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  c
>.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  B >.Cgr
<. e ,  f >.  /\  <. B ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. B ,  d >.Cgr <. f ,  h >. ) ) ) )
22 opeq2 3797 . . . . 5  |-  ( c  =  C  ->  <. A , 
c >.  =  <. A ,  C >. )
2322breq2d 4035 . . . 4  |-  ( c  =  C  ->  ( B  Btwn  <. A ,  c
>. 
<->  B  Btwn  <. A ,  C >. ) )
2423anbi1d 685 . . 3  |-  ( c  =  C  ->  (
( B  Btwn  <. A , 
c >.  /\  f  Btwn  <.
e ,  g >.
)  <->  ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. ) ) )
25 opeq2 3797 . . . . 5  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2625breq1d 4033 . . . 4  |-  ( c  =  C  ->  ( <. B ,  c >.Cgr <. f ,  g >.  <->  <. B ,  C >.Cgr <.
f ,  g >.
) )
2726anbi2d 684 . . 3  |-  ( c  =  C  ->  (
( <. A ,  B >.Cgr
<. e ,  f >.  /\  <. B ,  c
>.Cgr <. f ,  g
>. )  <->  ( <. A ,  B >.Cgr <. e ,  f
>.  /\  <. B ,  C >.Cgr
<. f ,  g >.
) ) )
2824, 273anbi12d 1253 . 2  |-  ( c  =  C  ->  (
( ( B  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  B >.Cgr <.
e ,  f >.  /\  <. B ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. B ,  d >.Cgr <. f ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  B >.Cgr
<. e ,  f >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\  <. B , 
d >.Cgr <. f ,  h >. ) ) ) )
29 opeq2 3797 . . . . 5  |-  ( d  =  D  ->  <. A , 
d >.  =  <. A ,  D >. )
3029breq1d 4033 . . . 4  |-  ( d  =  D  ->  ( <. A ,  d >.Cgr <. e ,  h >.  <->  <. A ,  D >.Cgr <. e ,  h >. ) )
31 opeq2 3797 . . . . 5  |-  ( d  =  D  ->  <. B , 
d >.  =  <. B ,  D >. )
3231breq1d 4033 . . . 4  |-  ( d  =  D  ->  ( <. B ,  d >.Cgr <. f ,  h >.  <->  <. B ,  D >.Cgr <. f ,  h >. ) )
3330, 32anbi12d 691 . . 3  |-  ( d  =  D  ->  (
( <. A ,  d
>.Cgr <. e ,  h >.  /\  <. B ,  d
>.Cgr <. f ,  h >. )  <->  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. B ,  D >.Cgr
<. f ,  h >. ) ) )
34333anbi3d 1258 . 2  |-  ( d  =  D  ->  (
( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  B >.Cgr <.
e ,  f >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\  <. B , 
d >.Cgr <. f ,  h >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  B >.Cgr
<. e ,  f >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. B ,  D >.Cgr <. f ,  h >. ) ) ) )
35 opeq1 3796 . . . . 5  |-  ( e  =  E  ->  <. e ,  g >.  =  <. E ,  g >. )
3635breq2d 4035 . . . 4  |-  ( e  =  E  ->  (
f  Btwn  <. e ,  g >.  <->  f  Btwn  <. E , 
g >. ) )
3736anbi2d 684 . . 3  |-  ( e  =  E  ->  (
( B  Btwn  <. A ,  C >.  /\  f  Btwn  <.
e ,  g >.
)  <->  ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E , 
g >. ) ) )
38 opeq1 3796 . . . . 5  |-  ( e  =  E  ->  <. e ,  f >.  =  <. E ,  f >. )
3938breq2d 4035 . . . 4  |-  ( e  =  E  ->  ( <. A ,  B >.Cgr <.
e ,  f >.  <->  <. A ,  B >.Cgr <. E ,  f >. ) )
4039anbi1d 685 . . 3  |-  ( e  =  E  ->  (
( <. A ,  B >.Cgr
<. e ,  f >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  <->  ( <. A ,  B >.Cgr <. E ,  f
>.  /\  <. B ,  C >.Cgr
<. f ,  g >.
) ) )
41 opeq1 3796 . . . . 5  |-  ( e  =  E  ->  <. e ,  h >.  =  <. E ,  h >. )
4241breq2d 4035 . . . 4  |-  ( e  =  E  ->  ( <. A ,  D >.Cgr <.
e ,  h >.  <->  <. A ,  D >.Cgr <. E ,  h >. ) )
4342anbi1d 685 . . 3  |-  ( e  =  E  ->  (
( <. A ,  D >.Cgr
<. e ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. ) ) )
4437, 40, 433anbi123d 1252 . 2  |-  ( e  =  E  ->  (
( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  B >.Cgr <.
e ,  f >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. B ,  D >.Cgr <. f ,  h >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E , 
g >. )  /\  ( <. A ,  B >.Cgr <. E ,  f >.  /\ 
<. B ,  C >.Cgr <.
f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr <. f ,  h >. ) ) ) )
45 breq1 4026 . . . 4  |-  ( f  =  F  ->  (
f  Btwn  <. E , 
g >. 
<->  F  Btwn  <. E , 
g >. ) )
4645anbi2d 684 . . 3  |-  ( f  =  F  ->  (
( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E ,  g >. )  <-> 
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  g >. ) ) )
47 opeq2 3797 . . . . 5  |-  ( f  =  F  ->  <. E , 
f >.  =  <. E ,  F >. )
4847breq2d 4035 . . . 4  |-  ( f  =  F  ->  ( <. A ,  B >.Cgr <. E ,  f >.  <->  <. A ,  B >.Cgr <. E ,  F >. ) )
49 opeq1 3796 . . . . 5  |-  ( f  =  F  ->  <. f ,  g >.  =  <. F ,  g >. )
5049breq2d 4035 . . . 4  |-  ( f  =  F  ->  ( <. B ,  C >.Cgr <.
f ,  g >.  <->  <. B ,  C >.Cgr <. F ,  g >. ) )
5148, 50anbi12d 691 . . 3  |-  ( f  =  F  ->  (
( <. A ,  B >.Cgr
<. E ,  f >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  <->  ( <. A ,  B >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
<. F ,  g >.
) ) )
52 opeq1 3796 . . . . 5  |-  ( f  =  F  ->  <. f ,  h >.  =  <. F ,  h >. )
5352breq2d 4035 . . . 4  |-  ( f  =  F  ->  ( <. B ,  D >.Cgr <.
f ,  h >.  <->  <. B ,  D >.Cgr <. F ,  h >. ) )
5453anbi2d 684 . . 3  |-  ( f  =  F  ->  (
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <. F ,  h >. ) ) )
5546, 51, 543anbi123d 1252 . 2  |-  ( f  =  F  ->  (
( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E , 
g >. )  /\  ( <. A ,  B >.Cgr <. E ,  f >.  /\ 
<. B ,  C >.Cgr <.
f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr <. f ,  h >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E , 
g >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  g >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr
<. F ,  h >. ) ) ) )
56 opeq2 3797 . . . . 5  |-  ( g  =  G  ->  <. E , 
g >.  =  <. E ,  G >. )
5756breq2d 4035 . . . 4  |-  ( g  =  G  ->  ( F  Btwn  <. E ,  g
>. 
<->  F  Btwn  <. E ,  G >. ) )
5857anbi2d 684 . . 3  |-  ( g  =  G  ->  (
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  g >. )  <-> 
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. ) ) )
59 opeq2 3797 . . . . 5  |-  ( g  =  G  ->  <. F , 
g >.  =  <. F ,  G >. )
6059breq2d 4035 . . . 4  |-  ( g  =  G  ->  ( <. B ,  C >.Cgr <. F ,  g >.  <->  <. B ,  C >.Cgr <. F ,  G >. ) )
6160anbi2d 684 . . 3  |-  ( g  =  G  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  g >. )  <-> 
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. ) ) )
6258, 613anbi12d 1253 . 2  |-  ( g  =  G  ->  (
( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E , 
g >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  g >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr
<. F ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr
<. F ,  h >. ) ) ) )
63 opeq2 3797 . . . . 5  |-  ( h  =  H  ->  <. E ,  h >.  =  <. E ,  H >. )
6463breq2d 4035 . . . 4  |-  ( h  =  H  ->  ( <. A ,  D >.Cgr <. E ,  h >.  <->  <. A ,  D >.Cgr <. E ,  H >. ) )
65 opeq2 3797 . . . . 5  |-  ( h  =  H  ->  <. F ,  h >.  =  <. F ,  H >. )
6665breq2d 4035 . . . 4  |-  ( h  =  H  ->  ( <. B ,  D >.Cgr <. F ,  h >.  <->  <. B ,  D >.Cgr <. F ,  H >. ) )
6764, 66anbi12d 691 . . 3  |-  ( h  =  H  ->  (
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <. F ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) ) )
68673anbi3d 1258 . 2  |-  ( h  =  H  ->  (
( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr
<. F ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
<. F ,  H >. ) ) ) )
69 fveq2 5525 . 2  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
70 df-ofs 24606 . 2  |-  OuterFiveSeg  =  { <. p ,  q >.  |  E. n  e.  NN  E. a  e.  ( EE `  n
) E. b  e.  ( EE `  n
) E. c  e.  ( EE `  n
) E. d  e.  ( EE `  n
) E. e  e.  ( EE `  n
) E. f  e.  ( EE `  n
) E. g  e.  ( EE `  n
) E. h  e.  ( EE `  n
) ( p  = 
<. <. a ,  b
>. ,  <. c ,  d >. >.  /\  q  =  <. <. e ,  f
>. ,  <. g ,  h >. >.  /\  ( (
b  Btwn  <. a ,  c >.  /\  f  Btwn  <. e ,  g
>. )  /\  ( <. a ,  b >.Cgr <. e ,  f >.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. a ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) ) ) }
7110, 21, 28, 34, 44, 55, 62, 68, 69, 70br8 24113 1  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
<. F ,  H >. ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517  Cgrccgr 24518    OuterFiveSeg cofs 24605
This theorem is referenced by:  5segofs  24629  ofscom  24630  cgrextend  24631  segconeq  24633  ifscgr  24667  brofs2  24700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ofs 24606
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