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Theorem restutopopn 18221
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 18216 . . . 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 18220 . . 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 18212 . . . . . . . . . . 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 18216 . . . . . . . . . 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 3318 . . . . . . 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 18190 . . . . . . . . . . . . 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 5247 . . . . . . . . . . . . 13  |-  ( ( t  i^i  w )
" { x }
)  C_  ( (
t " { x } )  i^i  (
w " { x } ) )
18 ssrin 3526 . . . . . . . . . . . . . . . 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 5161 . . . . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . . . . . . . . 20  |-  x  e. 
_V
27 inimasn 5248 . . . . . . . . . . . . . . . . . . . 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 3828 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  i^i  { x } )  =  (/)  <->  -.  x  e.  A )
3029bicomi 194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  x  e.  A  <->  ( A  i^i  { x } )  =  (/) )
3130necon1abii 2618 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  <->  x  e.  A )
32 xpima2 5274 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  i^i  { x } )  =/=  (/)  ->  (
( A  X.  A
) " { x } )  =  A )
3331, 32sylbir 205 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  A  ->  (
( A  X.  A
) " { x } )  =  A )
3433ineq2d 3502 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  A  ->  (
( w " {
x } )  i^i  ( ( A  X.  A ) " {
x } ) )  =  ( ( w
" { x }
)  i^i  A )
)
3528, 34syl5eq 2448 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  A  ->  (
( w  i^i  ( A  X.  A ) )
" { x }
)  =  ( ( w " { x } )  i^i  A
) )
36 incom 3493 . . . . . . . . . . . . . . . . . 18  |-  ( ( w " { x } )  i^i  A
)  =  ( A  i^i  ( w " { x } ) )
3735, 36syl6eq 2452 . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . 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 3345 . . . . . . . . . . . . . 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 3318 . . . . . . . . . . . . 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 3319 . . . . . . . . . . . 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 5157 . . . . . . . . . . . . . 14  |-  ( v  =  ( t  i^i  w )  ->  (
v " { x } )  =  ( ( t  i^i  w
) " { x } ) )
4544sseq1d 3335 . . . . . . . . . . . . 13  |-  ( v  =  ( t  i^i  w )  ->  (
( v " {
x } )  C_  b 
<->  ( ( t  i^i  w ) " {
x } )  C_  b ) )
4645rspcev 3012 . . . . . . . . . . . 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 2764 . . . . . . . . . . . 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 2810 . . . . . . . . . 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 4948 . . . . . . . . . . . . . 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 13610 . . . . . . . . . . . . 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 2810 . . . . . . . . 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 2764 . . . . . . . . 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 2810 . . . . . . . 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 2749 . . . . . . 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 18216 . . . . . . . 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 3294 . . . . . . . 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 2409 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  /\  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) )  ->  b  =  ( b  i^i 
A ) )
72 ineq1 3495 . . . . . . . 8  |-  ( a  =  b  ->  (
a  i^i  A )  =  ( b  i^i 
A ) )
7372eqeq2d 2415 . . . . . . 7  |-  ( a  =  b  ->  (
b  =  ( a  i^i  A )  <->  b  =  ( b  i^i  A
) ) )
7473rspcev 3012 . . . . . 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 5701 . . . . . . 7  |-  (unifTop `  U
)  e.  _V
77 elrest 13610 . . . . . . 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 3314 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
)  ->  (unifTop `  ( Ut  ( A  X.  A
) ) )  C_  ( (unifTop `  U )t  A
) )
834, 82eqssd 3325 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774    X. cxp 4835   "cima 4840   ` cfv 5413  (class class class)co 6040   ↾t crest 13603  UnifOncust 18182  unifTopcutop 18213
This theorem is referenced by:  ressusp  18248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-rest 13605  df-ust 18183  df-utop 18214
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