Step |
Hyp |
Ref |
Expression |
1 |
|
ioorf.1 |
|- F = ( x e. ran (,) |-> if ( x = (/) , <. 0 , 0 >. , <. inf ( x , RR* , < ) , sup ( x , RR* , < ) >. ) ) |
2 |
|
ioof |
|- (,) : ( RR* X. RR* ) --> ~P RR |
3 |
|
ffn |
|- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) ) |
4 |
|
ovelrn |
|- ( (,) Fn ( RR* X. RR* ) -> ( A e. ran (,) <-> E. a e. RR* E. b e. RR* A = ( a (,) b ) ) ) |
5 |
2 3 4
|
mp2b |
|- ( A e. ran (,) <-> E. a e. RR* E. b e. RR* A = ( a (,) b ) ) |
6 |
1
|
ioorinv2 |
|- ( ( a (,) b ) =/= (/) -> ( F ` ( a (,) b ) ) = <. a , b >. ) |
7 |
6
|
fveq2d |
|- ( ( a (,) b ) =/= (/) -> ( (,) ` ( F ` ( a (,) b ) ) ) = ( (,) ` <. a , b >. ) ) |
8 |
|
df-ov |
|- ( a (,) b ) = ( (,) ` <. a , b >. ) |
9 |
7 8
|
eqtr4di |
|- ( ( a (,) b ) =/= (/) -> ( (,) ` ( F ` ( a (,) b ) ) ) = ( a (,) b ) ) |
10 |
|
df-ne |
|- ( A =/= (/) <-> -. A = (/) ) |
11 |
|
neeq1 |
|- ( A = ( a (,) b ) -> ( A =/= (/) <-> ( a (,) b ) =/= (/) ) ) |
12 |
10 11
|
bitr3id |
|- ( A = ( a (,) b ) -> ( -. A = (/) <-> ( a (,) b ) =/= (/) ) ) |
13 |
|
2fveq3 |
|- ( A = ( a (,) b ) -> ( (,) ` ( F ` A ) ) = ( (,) ` ( F ` ( a (,) b ) ) ) ) |
14 |
|
id |
|- ( A = ( a (,) b ) -> A = ( a (,) b ) ) |
15 |
13 14
|
eqeq12d |
|- ( A = ( a (,) b ) -> ( ( (,) ` ( F ` A ) ) = A <-> ( (,) ` ( F ` ( a (,) b ) ) ) = ( a (,) b ) ) ) |
16 |
12 15
|
imbi12d |
|- ( A = ( a (,) b ) -> ( ( -. A = (/) -> ( (,) ` ( F ` A ) ) = A ) <-> ( ( a (,) b ) =/= (/) -> ( (,) ` ( F ` ( a (,) b ) ) ) = ( a (,) b ) ) ) ) |
17 |
9 16
|
mpbiri |
|- ( A = ( a (,) b ) -> ( -. A = (/) -> ( (,) ` ( F ` A ) ) = A ) ) |
18 |
17
|
a1i |
|- ( ( a e. RR* /\ b e. RR* ) -> ( A = ( a (,) b ) -> ( -. A = (/) -> ( (,) ` ( F ` A ) ) = A ) ) ) |
19 |
18
|
rexlimivv |
|- ( E. a e. RR* E. b e. RR* A = ( a (,) b ) -> ( -. A = (/) -> ( (,) ` ( F ` A ) ) = A ) ) |
20 |
5 19
|
sylbi |
|- ( A e. ran (,) -> ( -. A = (/) -> ( (,) ` ( F ` A ) ) = A ) ) |
21 |
|
ioorebas |
|- ( 0 (,) 0 ) e. ran (,) |
22 |
1
|
ioorval |
|- ( ( 0 (,) 0 ) e. ran (,) -> ( F ` ( 0 (,) 0 ) ) = if ( ( 0 (,) 0 ) = (/) , <. 0 , 0 >. , <. inf ( ( 0 (,) 0 ) , RR* , < ) , sup ( ( 0 (,) 0 ) , RR* , < ) >. ) ) |
23 |
21 22
|
ax-mp |
|- ( F ` ( 0 (,) 0 ) ) = if ( ( 0 (,) 0 ) = (/) , <. 0 , 0 >. , <. inf ( ( 0 (,) 0 ) , RR* , < ) , sup ( ( 0 (,) 0 ) , RR* , < ) >. ) |
24 |
|
iooid |
|- ( 0 (,) 0 ) = (/) |
25 |
24
|
iftruei |
|- if ( ( 0 (,) 0 ) = (/) , <. 0 , 0 >. , <. inf ( ( 0 (,) 0 ) , RR* , < ) , sup ( ( 0 (,) 0 ) , RR* , < ) >. ) = <. 0 , 0 >. |
26 |
23 25
|
eqtri |
|- ( F ` ( 0 (,) 0 ) ) = <. 0 , 0 >. |
27 |
26
|
fveq2i |
|- ( (,) ` ( F ` ( 0 (,) 0 ) ) ) = ( (,) ` <. 0 , 0 >. ) |
28 |
|
df-ov |
|- ( 0 (,) 0 ) = ( (,) ` <. 0 , 0 >. ) |
29 |
27 28
|
eqtr4i |
|- ( (,) ` ( F ` ( 0 (,) 0 ) ) ) = ( 0 (,) 0 ) |
30 |
24
|
eqeq2i |
|- ( A = ( 0 (,) 0 ) <-> A = (/) ) |
31 |
30
|
biimpri |
|- ( A = (/) -> A = ( 0 (,) 0 ) ) |
32 |
31
|
fveq2d |
|- ( A = (/) -> ( F ` A ) = ( F ` ( 0 (,) 0 ) ) ) |
33 |
32
|
fveq2d |
|- ( A = (/) -> ( (,) ` ( F ` A ) ) = ( (,) ` ( F ` ( 0 (,) 0 ) ) ) ) |
34 |
29 33 31
|
3eqtr4a |
|- ( A = (/) -> ( (,) ` ( F ` A ) ) = A ) |
35 |
20 34
|
pm2.61d2 |
|- ( A e. ran (,) -> ( (,) ` ( F ` A ) ) = A ) |