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Theorem restutopopn 18260
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 18255 . . . 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 607 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A  C_  X
)
3 restutop 18259 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
(unifTop `  U )t  A ) 
C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
42, 3syldan 457 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( (unifTop `  U )t  A )  C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
5 trust 18251 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
62, 5syldan 457 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( Ut  ( A  X.  A ) )  e.  (UnifOn `  A
) )
7 elutop 18255 . . . . . . . . . 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 607 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  A )
102adantr 452 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  A  C_  X )
119, 10sstrd 3350 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  X )
12 simp-9l 753 . . . . . . . . . . . . 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 732 . . . . . . . . . . . . 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 744 . . . . . . . . . . . . 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 18229 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  t  e.  U  /\  w  e.  U )  ->  (
t  i^i  w )  e.  U )
1612, 13, 14, 15syl3anc 1184 . . . . . . . . . . . 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 5280 . . . . . . . . . . . . 13  |-  ( ( t  i^i  w )
" { x }
)  C_  ( (
t " { x } )  i^i  (
w " { x } ) )
18 ssrin 3558 . . . . . . . . . . . . . . . 16  |-  ( ( t " { x } )  C_  A  ->  ( ( t " { x } )  i^i  ( w " { x } ) )  C_  ( A  i^i  ( w " {
x } ) ) )
1918adantl 453 . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . . 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 5194 . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . 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 3341 . . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . 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 2951 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
27 inimasn 5281 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  _V  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  (
( A  X.  A
) " { x } ) ) )
2826, 27ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  (
( A  X.  A
) " { x } ) )
29 disjsn 3860 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
3029bicomi 194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  x  e.  A  <->  ( A  i^i  { x } )  =  (/) )
3130necon1abii 2649 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  <->  x  e.  A )
32 xpima2 5307 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  ->  (
( A  X.  A
) " { x } )  =  A )
3331, 32sylbir 205 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  A  ->  (
( A  X.  A
) " { x } )  =  A )
3433ineq2d 3534 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  A  ->  (
( w " {
x } )  i^i  ( ( A  X.  A ) " {
x } ) )  =  ( ( w
" { x }
)  i^i  A )
)
3528, 34syl5eq 2479 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  A
) )
36 incom 3525 . . . . . . . . . . . . . . . . . 18  |-  ( ( w " { x } )  i^i  A
)  =  ( A  i^i  ( w " { x } ) )
3735, 36syl6eq 2483 . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 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 3377 . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . 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 3350 . . . . . . . . . . . . 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 3351 . . . . . . . . . . . 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 5190 . . . . . . . . . . . . . 14  |-  ( v  =  ( t  i^i  w )  ->  (
v " { x } )  =  ( ( t  i^i  w
) " { x } ) )
4544sseq1d 3367 . . . . . . . . . . . . 13  |-  ( v  =  ( t  i^i  w )  ->  (
( v " {
x } )  C_  b 
<->  ( ( t  i^i  w ) " {
x } )  C_  b ) )
4645rspcev 3044 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 743 . . . . . . . . . . . . 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 707 . . . . . . . . . . . 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 608 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A. x  e.  A  E. t  e.  U  ( t " { x } ) 
C_  A )
5150r19.21bi 2796 . . . . . . . . . . . 12  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  x  e.  A )  ->  E. t  e.  U  ( t " { x } ) 
C_  A )
5249, 24, 51syl2anc 643 . . . . . . . . . . 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 2842 . . . . . . . . . 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 732 . . . . . . . . . . 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 4981 . . . . . . . . . . . . . 14  |-  ( ( A  e.  (unifTop `  U
)  /\  A  e.  (unifTop `  U ) )  ->  ( A  X.  A )  e.  _V )
5655anidms 627 . . . . . . . . . . . . 13  |-  ( A  e.  (unifTop `  U
)  ->  ( A  X.  A )  e.  _V )
57 elrest 13647 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 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 471 . . . . . . . . . . 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 643 . . . . . . . . . 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 2842 . . . . . . . . 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 608 . . . . . . . . . 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 2796 . . . . . . . . 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 2842 . . . . . . . 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 2781 . . . . . . 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 18255 . . . . . . . 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 707 . . . . . . 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 889 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  (unifTop `  U )
)
69 df-ss 3326 . . . . . . . 8  |-  ( b 
C_  A  <->  ( b  i^i  A )  =  b )
709, 69sylib 189 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  (
b  i^i  A )  =  b )
7170eqcomd 2440 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  =  ( b  i^i 
A ) )
72 ineq1 3527 . . . . . . . 8  |-  ( a  =  b  ->  (
a  i^i  A )  =  ( b  i^i 
A ) )
7372eqeq2d 2446 . . . . . . 7  |-  ( a  =  b  ->  (
b  =  ( a  i^i  A )  <->  b  =  ( b  i^i  A
) ) )
7473rspcev 3044 . . . . . 6  |-  ( ( b  e.  (unifTop `  U
)  /\  b  =  ( b  i^i  A
) )  ->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) )
7568, 71, 74syl2anc 643 . . . . 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 5734 . . . . . . 7  |-  (unifTop `  U
)  e.  _V
77 elrest 13647 . . . . . . 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 652 . . . . . 6  |-  ( A  e.  (unifTop `  U
)  ->  ( b  e.  ( (unifTop `  U
)t 
A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
7978ad2antlr 708 . . . . 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 224 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  ( (unifTop `  U
)t 
A ) )
8180ex 424 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) )  ->  b  e.  ( (unifTop `  U )t  A
) ) )
8281ssrdv 3346 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  (unifTop `  ( Ut  ( A  X.  A
) ) )  C_  ( (unifTop `  U )t  A
) )
834, 82eqssd 3357 1  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( (unifTop `  U )t  A )  =  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806    X. cxp 4868   "cima 4873   ` cfv 5446  (class class class)co 6073   ↾t crest 13640  UnifOncust 18221  unifTopcutop 18252
This theorem is referenced by:  ressusp  18287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-rest 13642  df-ust 18222  df-utop 18253
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