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Theorem restutopopn 19818
Description: The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
restutopopn  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( (unifTop `  U )t  A )  =  (unifTop `  ( Ut  ( A  X.  A ) ) ) )

Proof of Theorem restutopopn
Dummy variables  a 
b  t  u  w  x  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elutop 19813 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  (unifTop `  U )  <->  ( A  C_  X  /\  A. x  e.  A  E. t  e.  U  (
t " { x } )  C_  A
) ) )
21simprbda 623 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A  C_  X
)
3 restutop 19817 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
(unifTop `  U )t  A ) 
C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
42, 3syldan 470 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( (unifTop `  U )t  A )  C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
5 trust 19809 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
62, 5syldan 470 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( Ut  ( A  X.  A ) )  e.  (UnifOn `  A
) )
7 elutop 19813 . . . . . . . . . 10  |-  ( ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A )  ->  (
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) )  <->  ( b  C_  A  /\  A. x  e.  b  E. u  e.  ( Ut  ( A  X.  A ) ) ( u " { x } )  C_  b
) ) )
86, 7syl 16 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) )  <->  ( b  C_  A  /\  A. x  e.  b  E. u  e.  ( Ut  ( A  X.  A ) ) ( u " { x } )  C_  b
) ) )
98simprbda 623 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  A )
102adantr 465 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  A  C_  X )
119, 10sstrd 3371 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  X )
12 simp-9l 775 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  ->  U  e.  (UnifOn `  X
) )
13 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
t  e.  U )
14 simp-4r 766 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  ->  w  e.  U )
15 ustincl 19787 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  t  e.  U  /\  w  e.  U )  ->  (
t  i^i  w )  e.  U )
1612, 13, 14, 15syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( t  i^i  w
)  e.  U )
17 inimass 5258 . . . . . . . . . . . . 13  |-  ( ( t  i^i  w )
" { x }
)  C_  ( (
t " { x } )  i^i  (
w " { x } ) )
18 ssrin 3580 . . . . . . . . . . . . . . . 16  |-  ( ( t " { x } )  C_  A  ->  ( ( t " { x } )  i^i  ( w " { x } ) )  C_  ( A  i^i  ( w " {
x } ) ) )
1918adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( ( t " { x } )  i^i  ( w " { x } ) )  C_  ( A  i^i  ( w " {
x } ) ) )
20 simpllr 758 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  ->  u  =  ( w  i^i  ( A  X.  A
) ) )
2120imaeq1d 5173 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( u " {
x } )  =  ( ( w  i^i  ( A  X.  A
) ) " {
x } ) )
229ad5antr 733 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  ->  b  C_  A
)
23 simp-5r 768 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  ->  x  e.  b )
2422, 23sseldd 3362 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  ->  x  e.  A
)
2524ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  ->  x  e.  A )
26 vex 2980 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
27 inimasn 5259 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  _V  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  (
( A  X.  A
) " { x } ) ) )
2826, 27ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  (
( A  X.  A
) " { x } ) )
29 disjsn 3941 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
3029bicomi 202 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  x  e.  A  <->  ( A  i^i  { x } )  =  (/) )
3130necon1abii 2667 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  <->  x  e.  A )
32 xpima2 5287 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  ->  (
( A  X.  A
) " { x } )  =  A )
3331, 32sylbir 213 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  A  ->  (
( A  X.  A
) " { x } )  =  A )
3433ineq2d 3557 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  A  ->  (
( w " {
x } )  i^i  ( ( A  X.  A ) " {
x } ) )  =  ( ( w
" { x }
)  i^i  A )
)
3528, 34syl5eq 2487 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  A
) )
36 incom 3548 . . . . . . . . . . . . . . . . . 18  |-  ( ( w " { x } )  i^i  A
)  =  ( A  i^i  ( w " { x } ) )
3735, 36syl6eq 2491 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( A  i^i  ( w " { x } ) ) )
3825, 37syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( ( w  i^i  ( A  X.  A
) ) " {
x } )  =  ( A  i^i  (
w " { x } ) ) )
3921, 38eqtrd 2475 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( u " {
x } )  =  ( A  i^i  (
w " { x } ) ) )
4019, 39sseqtr4d 3398 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( ( t " { x } )  i^i  ( w " { x } ) )  C_  ( u " { x } ) )
41 simp-5r 768 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( u " {
x } )  C_  b )
4240, 41sstrd 3371 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( ( t " { x } )  i^i  ( w " { x } ) )  C_  b )
4317, 42syl5ss 3372 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  -> 
( ( t  i^i  w ) " {
x } )  C_  b )
44 imaeq1 5169 . . . . . . . . . . . . . 14  |-  ( v  =  ( t  i^i  w )  ->  (
v " { x } )  =  ( ( t  i^i  w
) " { x } ) )
4544sseq1d 3388 . . . . . . . . . . . . 13  |-  ( v  =  ( t  i^i  w )  ->  (
( v " {
x } )  C_  b 
<->  ( ( t  i^i  w ) " {
x } )  C_  b ) )
4645rspcev 3078 . . . . . . . . . . . 12  |-  ( ( ( t  i^i  w
)  e.  U  /\  ( ( t  i^i  w ) " {
x } )  C_  b )  ->  E. v  e.  U  ( v " { x } ) 
C_  b )
4716, 43, 46syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  ->  E. v  e.  U  ( v " {
x } )  C_  b )
48 simp-4l 765 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  -> 
( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
) )
4948ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  ->  ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) ) )
501simplbda 624 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A. x  e.  A  E. t  e.  U  ( t " { x } ) 
C_  A )
5150r19.21bi 2819 . . . . . . . . . . . 12  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  x  e.  A )  ->  E. t  e.  U  ( t " { x } ) 
C_  A )
5249, 24, 51syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  ->  E. t  e.  U  ( t " {
x } )  C_  A )
5347, 52r19.29a 2867 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  ->  E. v  e.  U  ( v " {
x } )  C_  b )
54 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  ->  u  e.  ( Ut  ( A  X.  A ) ) )
55 xpexg 6512 . . . . . . . . . . . . . 14  |-  ( ( A  e.  (unifTop `  U
)  /\  A  e.  (unifTop `  U ) )  ->  ( A  X.  A )  e.  _V )
5655anidms 645 . . . . . . . . . . . . 13  |-  ( A  e.  (unifTop `  U
)  ->  ( A  X.  A )  e.  _V )
57 elrest 14371 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  ( A  X.  A )  e. 
_V )  ->  (
u  e.  ( Ut  ( A  X.  A ) )  <->  E. w  e.  U  u  =  ( w  i^i  ( A  X.  A
) ) ) )
5856, 57sylan2 474 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( u  e.  ( Ut  ( A  X.  A ) )  <->  E. w  e.  U  u  =  ( w  i^i  ( A  X.  A ) ) ) )
5958biimpa 484 . . . . . . . . . . 11  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  u  e.  ( Ut  ( A  X.  A ) ) )  ->  E. w  e.  U  u  =  ( w  i^i  ( A  X.  A
) ) )
6048, 54, 59syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  ->  E. w  e.  U  u  =  ( w  i^i  ( A  X.  A
) ) )
6153, 60r19.29a 2867 . . . . . . . . 9  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  /\  x  e.  b )  /\  u  e.  ( Ut  ( A  X.  A ) ) )  /\  ( u " { x } ) 
C_  b )  ->  E. v  e.  U  ( v " {
x } )  C_  b )
628simplbda 624 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  A. x  e.  b  E. u  e.  ( Ut  ( A  X.  A ) ) ( u " { x } )  C_  b
)
6362r19.21bi 2819 . . . . . . . . 9  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  ->  E. u  e.  ( Ut  ( A  X.  A ) ) ( u " { x } )  C_  b
)
6461, 63r19.29a 2867 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U ) )  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )  /\  x  e.  b )  ->  E. v  e.  U  ( v " { x } ) 
C_  b )
6564ralrimiva 2804 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  A. x  e.  b  E. v  e.  U  ( v " { x } ) 
C_  b )
66 elutop 19813 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  ( b  e.  (unifTop `  U )  <->  ( b  C_  X  /\  A. x  e.  b  E. v  e.  U  (
v " { x } )  C_  b
) ) )
6766ad2antrr 725 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  (
b  e.  (unifTop `  U
)  <->  ( b  C_  X  /\  A. x  e.  b  E. v  e.  U  ( v " { x } ) 
C_  b ) ) )
6811, 65, 67mpbir2and 913 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  (unifTop `  U )
)
69 df-ss 3347 . . . . . . . 8  |-  ( b 
C_  A  <->  ( b  i^i  A )  =  b )
709, 69sylib 196 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  (
b  i^i  A )  =  b )
7170eqcomd 2448 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  =  ( b  i^i 
A ) )
72 ineq1 3550 . . . . . . . 8  |-  ( a  =  b  ->  (
a  i^i  A )  =  ( b  i^i 
A ) )
7372eqeq2d 2454 . . . . . . 7  |-  ( a  =  b  ->  (
b  =  ( a  i^i  A )  <->  b  =  ( b  i^i  A
) ) )
7473rspcev 3078 . . . . . 6  |-  ( ( b  e.  (unifTop `  U
)  /\  b  =  ( b  i^i  A
) )  ->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) )
7568, 71, 74syl2anc 661 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) )
76 fvex 5706 . . . . . . 7  |-  (unifTop `  U
)  e.  _V
77 elrest 14371 . . . . . . 7  |-  ( ( (unifTop `  U )  e.  _V  /\  A  e.  (unifTop `  U )
)  ->  ( b  e.  ( (unifTop `  U
)t 
A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
7876, 77mpan 670 . . . . . 6  |-  ( A  e.  (unifTop `  U
)  ->  ( b  e.  ( (unifTop `  U
)t 
A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
7978ad2antlr 726 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  (
b  e.  ( (unifTop `  U )t  A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
8075, 79mpbird 232 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  ( (unifTop `  U
)t 
A ) )
8180ex 434 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) )  ->  b  e.  ( (unifTop `  U )t  A
) ) )
8281ssrdv 3367 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  (unifTop `  ( Ut  ( A  X.  A
) ) )  C_  ( (unifTop `  U )t  A
) )
834, 82eqssd 3378 1  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( (unifTop `  U )t  A )  =  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    i^i cin 3332    C_ wss 3333   (/)c0 3642   {csn 3882    X. cxp 4843   "cima 4848   ` cfv 5423  (class class class)co 6096   ↾t crest 14364  UnifOncust 19779  unifTopcutop 19810
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-rest 14366  df-ust 19780  df-utop 19811
This theorem is referenced by:  ressusp  19845
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