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Theorem restutopopn 19772
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 19767 . . . 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 620 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A  C_  X
)
3 restutop 19771 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
(unifTop `  U )t  A ) 
C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
42, 3syldan 467 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( (unifTop `  U )t  A )  C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
5 trust 19763 . . . . . . . . . . 11  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
62, 5syldan 467 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  ( Ut  ( A  X.  A ) )  e.  (UnifOn `  A
) )
7 elutop 19767 . . . . . . . . . 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 620 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  A )
102adantr 462 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  A  C_  X )
119, 10sstrd 3363 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  C_  X )
12 simp-9l 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 )  /\  w  e.  U )  /\  u  =  (
w  i^i  ( A  X.  A ) ) )  /\  t  e.  U
)  /\  ( t " { x } ) 
C_  A )  ->  U  e.  (UnifOn `  X
) )
13 simplr 749 . . . . . . . . . . . . 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 761 . . . . . . . . . . . . 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 19741 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  t  e.  U  /\  w  e.  U )  ->  (
t  i^i  w )  e.  U )
1612, 13, 14, 15syl3anc 1213 . . . . . . . . . . . 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 5250 . . . . . . . . . . . . 13  |-  ( ( t  i^i  w )
" { x }
)  C_  ( (
t " { x } )  i^i  (
w " { x } ) )
18 ssrin 3572 . . . . . . . . . . . . . . . 16  |-  ( ( t " { x } )  C_  A  ->  ( ( t " { x } )  i^i  ( w " { x } ) )  C_  ( A  i^i  ( w " {
x } ) ) )
1918adantl 463 . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . 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 5165 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . . . . 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 2973 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
27 inimasn 5251 . . . . . . . . . . . . . . . . . . . 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 3933 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
3029bicomi 202 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  x  e.  A  <->  ( A  i^i  { x } )  =  (/) )
3130necon1abii 2660 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  <->  x  e.  A )
32 xpima2 5279 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  ->  (
( A  X.  A
) " { x } )  =  A )
3331, 32sylbir 213 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  A  ->  (
( A  X.  A
) " { x } )  =  A )
3433ineq2d 3549 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  A  ->  (
( w " {
x } )  i^i  ( ( A  X.  A ) " {
x } ) )  =  ( ( w
" { x }
)  i^i  A )
)
3528, 34syl5eq 2485 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  A
) )
36 incom 3540 . . . . . . . . . . . . . . . . . 18  |-  ( ( w " { x } )  i^i  A
)  =  ( A  i^i  ( w " { x } ) )
3735, 36syl6eq 2489 . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . 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 3390 . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 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 3363 . . . . . . . . . . . . 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 3364 . . . . . . . . . . . 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 5161 . . . . . . . . . . . . . 14  |-  ( v  =  ( t  i^i  w )  ->  (
v " { x } )  =  ( ( t  i^i  w
) " { x } ) )
4544sseq1d 3380 . . . . . . . . . . . . 13  |-  ( v  =  ( t  i^i  w )  ->  (
( v " {
x } )  C_  b 
<->  ( ( t  i^i  w ) " {
x } )  C_  b ) )
4645rspcev 3070 . . . . . . . . . . . 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 656 . . . . . . . . . . 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 760 . . . . . . . . . . . . 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 720 . . . . . . . . . . . 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 621 . . . . . . . . . . . . 13  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  A. x  e.  A  E. t  e.  U  ( t " { x } ) 
C_  A )
5150r19.21bi 2812 . . . . . . . . . . . 12  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  x  e.  A )  ->  E. t  e.  U  ( t " { x } ) 
C_  A )
5249, 24, 51syl2anc 656 . . . . . . . . . . 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 2860 . . . . . . . . . 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 749 . . . . . . . . . . 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 6506 . . . . . . . . . . . . . 14  |-  ( ( A  e.  (unifTop `  U
)  /\  A  e.  (unifTop `  U ) )  ->  ( A  X.  A )  e.  _V )
5655anidms 640 . . . . . . . . . . . . 13  |-  ( A  e.  (unifTop `  U
)  ->  ( A  X.  A )  e.  _V )
57 elrest 14362 . . . . . . . . . . . . 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 471 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 656 . . . . . . . . . 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 2860 . . . . . . . . 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 621 . . . . . . . . . 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 2812 . . . . . . . . 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 2860 . . . . . . . 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 2797 . . . . . . 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 19767 . . . . . . . 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 720 . . . . . . 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 908 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  e.  (unifTop `  U )
)
69 df-ss 3339 . . . . . . . 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 2446 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  =  ( b  i^i 
A ) )
72 ineq1 3542 . . . . . . . 8  |-  ( a  =  b  ->  (
a  i^i  A )  =  ( b  i^i 
A ) )
7372eqeq2d 2452 . . . . . . 7  |-  ( a  =  b  ->  (
b  =  ( a  i^i  A )  <->  b  =  ( b  i^i  A
) ) )
7473rspcev 3070 . . . . . 6  |-  ( ( b  e.  (unifTop `  U
)  /\  b  =  ( b  i^i  A
) )  ->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) )
7568, 71, 74syl2anc 656 . . . . 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 5698 . . . . . . 7  |-  (unifTop `  U
)  e.  _V
77 elrest 14362 . . . . . . 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 665 . . . . . 6  |-  ( A  e.  (unifTop `  U
)  ->  ( b  e.  ( (unifTop `  U
)t 
A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
7978ad2antlr 721 . . . . 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 3359 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  (unifTop `  ( Ut  ( A  X.  A
) ) )  C_  ( (unifTop `  U )t  A
) )
834, 82eqssd 3370 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874    X. cxp 4834   "cima 4839   ` cfv 5415  (class class class)co 6090   ↾t crest 14355  UnifOncust 19733  unifTopcutop 19764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-rest 14357  df-ust 19734  df-utop 19765
This theorem is referenced by:  ressusp  19799
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