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Theorem restutop 20567
Description: Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
Assertion
Ref Expression
restutop  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
(unifTop `  U )t  A ) 
C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )

Proof of Theorem restutop
Dummy variables  a 
b  u  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
( U  e.  (UnifOn `  X )  /\  A  C_  X ) )
2 fvex 5876 . . . . . . . 8  |-  (unifTop `  U
)  e.  _V
32a1i 11 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (unifTop `  U )  e.  _V )
4 elfvex 5893 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
54adantr 465 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
6 simpr 461 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
75, 6ssexd 4594 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
8 elrest 14686 . . . . . . 7  |-  ( ( (unifTop `  U )  e.  _V  /\  A  e. 
_V )  ->  (
b  e.  ( (unifTop `  U )t  A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
93, 7, 8syl2anc 661 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
b  e.  ( (unifTop `  U )t  A )  <->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) ) )
109biimpa 484 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  ->  E. a  e.  (unifTop `  U ) b  =  ( a  i^i  A
) )
11 inss2 3719 . . . . . . 7  |-  ( a  i^i  A )  C_  A
12 sseq1 3525 . . . . . . 7  |-  ( b  =  ( a  i^i 
A )  ->  (
b  C_  A  <->  ( a  i^i  A )  C_  A
) )
1311, 12mpbiri 233 . . . . . 6  |-  ( b  =  ( a  i^i 
A )  ->  b  C_  A )
1413rexlimivw 2952 . . . . 5  |-  ( E. a  e.  (unifTop `  U
) b  =  ( a  i^i  A )  ->  b  C_  A
)
1510, 14syl 16 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
b  C_  A )
16 simp-5l 767 . . . . . . . . . 10  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  ->  U  e.  (UnifOn `  X
) )
1716ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  U  e.  (UnifOn `  X ) )
187ad6antr 735 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  A  e.  _V )
19 xpexg 6587 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2018, 18, 19syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  ( A  X.  A )  e.  _V )
21 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  u  e.  U )
22 elrestr 14687 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  ( A  X.  A )  e. 
_V  /\  u  e.  U )  ->  (
u  i^i  ( A  X.  A ) )  e.  ( Ut  ( A  X.  A ) ) )
2317, 20, 21, 22syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  ( u  i^i  ( A  X.  A
) )  e.  ( Ut  ( A  X.  A
) ) )
24 inss1 3718 . . . . . . . . . . . . 13  |-  ( u  i^i  ( A  X.  A ) )  C_  u
25 imass1 5371 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) ) 
C_  u  ->  (
( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( u " { x } ) )
2624, 25ax-mp 5 . . . . . . . . . . . 12  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( u " { x } )
27 sstr 3512 . . . . . . . . . . . 12  |-  ( ( ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  ( u " {
x } )  /\  ( u " {
x } )  C_  a )  ->  (
( u  i^i  ( A  X.  A ) )
" { x }
)  C_  a )
2826, 27mpan 670 . . . . . . . . . . 11  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  a )
29 imassrn 5348 . . . . . . . . . . . . . . 15  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( u  i^i  ( A  X.  A ) )
30 rnin 5415 . . . . . . . . . . . . . . 15  |-  ran  (
u  i^i  ( A  X.  A ) )  C_  ( ran  u  i^i  ran  ( A  X.  A
) )
3129, 30sstri 3513 . . . . . . . . . . . . . 14  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( ran  u  i^i  ran  ( A  X.  A ) )
32 inss2 3719 . . . . . . . . . . . . . 14  |-  ( ran  u  i^i  ran  ( A  X.  A ) ) 
C_  ran  ( A  X.  A )
3331, 32sstri 3513 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( A  X.  A )
34 rnxpid 5440 . . . . . . . . . . . . 13  |-  ran  ( A  X.  A )  =  A
3533, 34sseqtri 3536 . . . . . . . . . . . 12  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  A
3635a1i 11 . . . . . . . . . . 11  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  A )
3728, 36ssind 3722 . . . . . . . . . 10  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  ( a  i^i  A
) )
3837adantl 466 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  ( (
u  i^i  ( A  X.  A ) ) " { x } ) 
C_  ( a  i^i 
A ) )
39 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  b  =  ( a  i^i  A
) )
4038, 39sseqtr4d 3541 . . . . . . . 8  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  ( (
u  i^i  ( A  X.  A ) ) " { x } ) 
C_  b )
41 imaeq1 5332 . . . . . . . . . 10  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
v " { x } )  =  ( ( u  i^i  ( A  X.  A ) )
" { x }
) )
4241sseq1d 3531 . . . . . . . . 9  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
( v " {
x } )  C_  b 
<->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  b ) )
4342rspcev 3214 . . . . . . . 8  |-  ( ( ( u  i^i  ( A  X.  A ) )  e.  ( Ut  ( A  X.  A ) )  /\  ( ( u  i^i  ( A  X.  A ) ) " { x } ) 
C_  b )  ->  E. v  e.  ( Ut  ( A  X.  A
) ) ( v
" { x }
)  C_  b )
4423, 40, 43syl2anc 661 . . . . . . 7  |-  ( ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U )
)  /\  b  =  ( a  i^i  A
) )  /\  u  e.  U )  /\  (
u " { x } )  C_  a
)  ->  E. v  e.  ( Ut  ( A  X.  A ) ) ( v " { x } )  C_  b
)
45 simplr 754 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  -> 
a  e.  (unifTop `  U
) )
46 inss1 3718 . . . . . . . . 9  |-  ( a  i^i  A )  C_  a
47 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  ->  x  e.  b )
48 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  -> 
b  =  ( a  i^i  A ) )
4947, 48eleqtrd 2557 . . . . . . . . 9  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  ->  x  e.  ( a  i^i  A ) )
5046, 49sseldi 3502 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  ->  x  e.  a )
51 elutop 20563 . . . . . . . . . 10  |-  ( U  e.  (UnifOn `  X
)  ->  ( a  e.  (unifTop `  U )  <->  ( a  C_  X  /\  A. x  e.  a  E. u  e.  U  (
u " { x } )  C_  a
) ) )
5251simplbda 624 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  ->  A. x  e.  a  E. u  e.  U  ( u " { x } ) 
C_  a )
5352r19.21bi 2833 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  /\  x  e.  a )  ->  E. u  e.  U  ( u " { x } ) 
C_  a )
5416, 45, 50, 53syl21anc 1227 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  ->  E. u  e.  U  ( u " {
x } )  C_  a )
5544, 54r19.29a 3003 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  /\  x  e.  b )  /\  a  e.  (unifTop `  U ) )  /\  b  =  ( a  i^i  A ) )  ->  E. v  e.  ( Ut  ( A  X.  A
) ) ( v
" { x }
)  C_  b )
5610adantr 465 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  ->  E. a  e.  (unifTop `  U )
b  =  ( a  i^i  A ) )
5755, 56r19.29a 3003 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U )t  A
) )  /\  x  e.  b )  ->  E. v  e.  ( Ut  ( A  X.  A ) ) ( v " { x } )  C_  b
)
5857ralrimiva 2878 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  ->  A. x  e.  b  E. v  e.  ( Ut  ( A  X.  A
) ) ( v
" { x }
)  C_  b )
59 trust 20559 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
60 elutop 20563 . . . . . 6  |-  ( ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A )  ->  (
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) )  <->  ( b  C_  A  /\  A. x  e.  b  E. v  e.  ( Ut  ( A  X.  A ) ) ( v " { x } )  C_  b
) ) )
6159, 60syl 16 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) )  <->  ( b  C_  A  /\  A. x  e.  b  E. v  e.  ( Ut  ( A  X.  A ) ) ( v " { x } )  C_  b
) ) )
6261biimpar 485 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  (
b  C_  A  /\  A. x  e.  b  E. v  e.  ( Ut  ( A  X.  A ) ) ( v " {
x } )  C_  b ) )  -> 
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )
631, 15, 58, 62syl12anc 1226 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )
6463ex 434 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
b  e.  ( (unifTop `  U )t  A )  ->  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) ) )
6564ssrdv 3510 1  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
(unifTop `  U )t  A ) 
C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   {csn 4027    X. cxp 4997   ran crn 5000   "cima 5002   ` cfv 5588  (class class class)co 6285   ↾t crest 14679  UnifOncust 20529  unifTopcutop 20560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-rest 14681  df-ust 20530  df-utop 20561
This theorem is referenced by:  restutopopn  20568
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