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Theorem restutop 18259
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 444 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
( U  e.  (UnifOn `  X )  /\  A  C_  X ) )
2 fvex 5734 . . . . . . . 8  |-  (unifTop `  U
)  e.  _V
32a1i 11 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (unifTop `  U )  e.  _V )
4 elfvex 5750 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
54adantr 452 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
6 simpr 448 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
75, 6ssexd 4342 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
8 elrest 13647 . . . . . . 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 643 . . . . . 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 471 . . . . 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 3554 . . . . . . 7  |-  ( a  i^i  A )  C_  A
12 sseq1 3361 . . . . . . 7  |-  ( b  =  ( a  i^i 
A )  ->  (
b  C_  A  <->  ( a  i^i  A )  C_  A
) )
1311, 12mpbiri 225 . . . . . 6  |-  ( b  =  ( a  i^i 
A )  ->  b  C_  A )
1413rexlimivw 2818 . . . . 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 745 . . . . . . . . . 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 707 . . . . . . . . 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 717 . . . . . . . . . 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 4981 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2018, 18, 19syl2anc 643 . . . . . . . . 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 732 . . . . . . . . 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 13648 . . . . . . . . 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 1184 . . . . . . . 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 3553 . . . . . . . . . . . . 13  |-  ( u  i^i  ( A  X.  A ) )  C_  u
25 imass1 5231 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) ) 
C_  u  ->  (
( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( u " { x } ) )
2624, 25ax-mp 8 . . . . . . . . . . . 12  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( u " { x } )
27 sstr 3348 . . . . . . . . . . . 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 652 . . . . . . . . . . 11  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  a )
29 imassrn 5208 . . . . . . . . . . . . . . 15  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( u  i^i  ( A  X.  A ) )
30 rnin 5273 . . . . . . . . . . . . . . 15  |-  ran  (
u  i^i  ( A  X.  A ) )  C_  ( ran  u  i^i  ran  ( A  X.  A
) )
3129, 30sstri 3349 . . . . . . . . . . . . . 14  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( ran  u  i^i  ran  ( A  X.  A ) )
32 inss2 3554 . . . . . . . . . . . . . 14  |-  ( ran  u  i^i  ran  ( A  X.  A ) ) 
C_  ran  ( A  X.  A )
3331, 32sstri 3349 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( A  X.  A )
34 rnxpid 5294 . . . . . . . . . . . . 13  |-  ran  ( A  X.  A )  =  A
3533, 34sseqtri 3372 . . . . . . . . . . . 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 3557 . . . . . . . . . 10  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  ( a  i^i  A
) )
3837adantl 453 . . . . . . . . 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 736 . . . . . . . . 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 3377 . . . . . . . 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 5190 . . . . . . . . . 10  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
v " { x } )  =  ( ( u  i^i  ( A  X.  A ) )
" { x }
) )
4241sseq1d 3367 . . . . . . . . 9  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
( v " {
x } )  C_  b 
<->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  b ) )
4342rspcev 3044 . . . . . . . 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 643 . . . . . . 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 732 . . . . . . . 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 3553 . . . . . . . . 9  |-  ( a  i^i  A )  C_  a
47 simpllr 736 . . . . . . . . . 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 448 . . . . . . . . . 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 2511 . . . . . . . . 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 3338 . . . . . . . 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 18255 . . . . . . . . . 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 608 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  ->  A. x  e.  a  E. u  e.  U  ( u " { x } ) 
C_  a )
5352r19.21bi 2796 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  /\  x  e.  a )  ->  E. u  e.  U  ( u " { x } ) 
C_  a )
5416, 45, 50, 53syl21anc 1183 . . . . . . 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 2842 . . . . . 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 452 . . . . . 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 2842 . . . . 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 2781 . . . 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 18251 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
60 elutop 18255 . . . . . 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 472 . . . 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 1182 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )
6463ex 424 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
b  e.  ( (unifTop `  U )t  A )  ->  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) ) )
6564ssrdv 3346 1  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
(unifTop `  U )t  A ) 
C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {csn 3806    X. cxp 4868   ran crn 4871   "cima 4873   ` cfv 5446  (class class class)co 6073   ↾t crest 13640  UnifOncust 18221  unifTopcutop 18252
This theorem is referenced by:  restutopopn  18260
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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