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Theorem restutopopn 21035
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 21030 . . . 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 21034 . . 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 21026 . . . . . . . . . . 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 21030 . . . . . . . . . 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 17 . . . . . . . . 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 3454 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  X )
12 simp-9l 780 . . . . . . . . . . . . 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 756 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 21004 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  t  e.  U  /\  w  e.  U )  ->  (
t  i^i  w )  e.  U )
1612, 13, 14, 15syl3anc 1232 . . . . . . . . . . . 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 5242 . . . . . . . . . . . . 13  |-  ( ( t  i^i  w )
" { x }
)  C_  ( (
t " { x } )  i^i  (
w " { x } ) )
18 ssrin 3666 . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . 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 5158 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . 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 3445 . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . 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 3064 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
27 inimasn 5243 . . . . . . . . . . . . . . . . . . . 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 xpimasn 5272 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  A  ->  (
( A  X.  A
) " { x } )  =  A )
3029ineq2d 3643 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  A  ->  (
( w " {
x } )  i^i  ( ( A  X.  A ) " {
x } ) )  =  ( ( w
" { x }
)  i^i  A )
)
3128, 30syl5eq 2457 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  A
) )
32 incom 3634 . . . . . . . . . . . . . . . . . 18  |-  ( ( w " { x } )  i^i  A
)  =  ( A  i^i  ( w " { x } ) )
3331, 32syl6eq 2461 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( A  i^i  ( w " { x } ) ) )
3425, 33syl 17 . . . . . . . . . . . . . . . 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 } ) ) )
3521, 34eqtrd 2445 . . . . . . . . . . . . . . 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 } ) ) )
3619, 35sseqtr4d 3481 . . . . . . . . . . . . . 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 } ) )
37 simp-5r 773 . . . . . . . . . . . . . 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 )
3836, 37sstrd 3454 . . . . . . . . . . . . 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 )
3917, 38syl5ss 3455 . . . . . . . . . . . 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 )
40 imaeq1 5154 . . . . . . . . . . . . . 14  |-  ( v  =  ( t  i^i  w )  ->  (
v " { x } )  =  ( ( t  i^i  w
) " { x } ) )
4140sseq1d 3471 . . . . . . . . . . . . 13  |-  ( v  =  ( t  i^i  w )  ->  (
( v " {
x } )  C_  b 
<->  ( ( t  i^i  w ) " {
x } )  C_  b ) )
4241rspcev 3162 . . . . . . . . . . . 12  |-  ( ( ( t  i^i  w
)  e.  U  /\  ( ( t  i^i  w ) " {
x } )  C_  b )  ->  E. v  e.  U  ( v " { x } ) 
C_  b )
4316, 39, 42syl2anc 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 )
44 simp-4l 770 . . . . . . . . . . . . 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 )
) )
4544ad2antrr 726 . . . . . . . . . . . 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 ) ) )
461simplbda 624 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A. x  e.  A  E. t  e.  U  ( t " { x } ) 
C_  A )
4746r19.21bi 2775 . . . . . . . . . . . 12  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  x  e.  A )  ->  E. t  e.  U  ( t " { x } ) 
C_  A )
4845, 24, 47syl2anc 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 )
4943, 48r19.29a 2951 . . . . . . . . . 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 )
50 simplr 756 . . . . . . . . . . 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 ) ) )
51 sqxpexg 6589 . . . . . . . . . . . . 13  |-  ( A  e.  (unifTop `  U
)  ->  ( A  X.  A )  e.  _V )
52 elrest 15044 . . . . . . . . . . . . 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
) ) ) )
5351, 52sylan2 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 ) ) ) )
5453biimpa 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
) ) )
5544, 50, 54syl2anc 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
) ) )
5649, 55r19.29a 2951 . . . . . . . . 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 )
578simplbda 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
)
5857r19.21bi 2775 . . . . . . . . 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
)
5956, 58r19.29a 2951 . . . . . . . 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 )
6059ralrimiva 2820 . . . . . . 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 )
61 elutop 21030 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  ( b  e.  (unifTop `  U )  <->  ( b  C_  X  /\  A. x  e.  b  E. v  e.  U  (
v " { x } )  C_  b
) ) )
6261ad2antrr 726 . . . . . . 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 ) ) )
6311, 60, 62mpbir2and 925 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  (unifTop `  U )
)
64 df-ss 3430 . . . . . . . 8  |-  ( b 
C_  A  <->  ( b  i^i  A )  =  b )
659, 64sylib 198 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  (
b  i^i  A )  =  b )
6665eqcomd 2412 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  =  ( b  i^i 
A ) )
67 ineq1 3636 . . . . . . . 8  |-  ( a  =  b  ->  (
a  i^i  A )  =  ( b  i^i 
A ) )
6867eqeq2d 2418 . . . . . . 7  |-  ( a  =  b  ->  (
b  =  ( a  i^i  A )  <->  b  =  ( b  i^i  A
) ) )
6968rspcev 3162 . . . . . 6  |-  ( ( b  e.  (unifTop `  U
)  /\  b  =  ( b  i^i  A
) )  ->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) )
7063, 66, 69syl2anc 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 ) )
71 fvex 5861 . . . . . . 7  |-  (unifTop `  U
)  e.  _V
72 elrest 15044 . . . . . . 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 ) ) )
7371, 72mpan 670 . . . . . 6  |-  ( A  e.  (unifTop `  U
)  ->  ( b  e.  ( (unifTop `  U
)t 
A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
7473ad2antlr 727 . . . . 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 ) ) )
7570, 74mpbird 234 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  ( (unifTop `  U
)t 
A ) )
7675ex 434 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) )  ->  b  e.  ( (unifTop `  U )t  A
) ) )
7776ssrdv 3450 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  (unifTop `  ( Ut  ( A  X.  A
) ) )  C_  ( (unifTop `  U )t  A
) )
784, 77eqssd 3461 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:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   _Vcvv 3061    i^i cin 3415    C_ wss 3416   {csn 3974    X. cxp 4823   "cima 4828   ` cfv 5571  (class class class)co 6280   ↾t crest 15037  UnifOncust 20996  unifTopcutop 21027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-rest 15039  df-ust 20997  df-utop 21028
This theorem is referenced by:  ressusp  21062
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