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Theorem restutop 19771
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 454 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
( U  e.  (UnifOn `  X )  /\  A  C_  X ) )
2 fvex 5698 . . . . . . . 8  |-  (unifTop `  U
)  e.  _V
32a1i 11 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (unifTop `  U )  e.  _V )
4 elfvex 5714 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
54adantr 462 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
6 simpr 458 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
75, 6ssexd 4436 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
8 elrest 14362 . . . . . . 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 656 . . . . . 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 481 . . . . 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 3568 . . . . . . 7  |-  ( a  i^i  A )  C_  A
12 sseq1 3374 . . . . . . 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 2835 . . . . 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 762 . . . . . . . . . 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 720 . . . . . . . . 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 730 . . . . . . . . . 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 6506 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2018, 18, 19syl2anc 656 . . . . . . . . 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 749 . . . . . . . . 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 14363 . . . . . . . . 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 1213 . . . . . . . 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 3567 . . . . . . . . . . . . 13  |-  ( u  i^i  ( A  X.  A ) )  C_  u
25 imass1 5200 . . . . . . . . . . . . 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 3361 . . . . . . . . . . . 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 665 . . . . . . . . . . 11  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  a )
29 imassrn 5177 . . . . . . . . . . . . . . 15  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( u  i^i  ( A  X.  A ) )
30 rnin 5243 . . . . . . . . . . . . . . 15  |-  ran  (
u  i^i  ( A  X.  A ) )  C_  ( ran  u  i^i  ran  ( A  X.  A
) )
3129, 30sstri 3362 . . . . . . . . . . . . . 14  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( ran  u  i^i  ran  ( A  X.  A ) )
32 inss2 3568 . . . . . . . . . . . . . 14  |-  ( ran  u  i^i  ran  ( A  X.  A ) ) 
C_  ran  ( A  X.  A )
3331, 32sstri 3362 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( A  X.  A )
34 rnxpid 5268 . . . . . . . . . . . . 13  |-  ran  ( A  X.  A )  =  A
3533, 34sseqtri 3385 . . . . . . . . . . . 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 3571 . . . . . . . . . 10  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  ( a  i^i  A
) )
3837adantl 463 . . . . . . . . 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 753 . . . . . . . . 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 3390 . . . . . . . 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 5161 . . . . . . . . . 10  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
v " { x } )  =  ( ( u  i^i  ( A  X.  A ) )
" { x }
) )
4241sseq1d 3380 . . . . . . . . 9  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
( v " {
x } )  C_  b 
<->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  b ) )
4342rspcev 3070 . . . . . . . 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 656 . . . . . . 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 749 . . . . . . . 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 3567 . . . . . . . . 9  |-  ( a  i^i  A )  C_  a
47 simpllr 753 . . . . . . . . . 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 458 . . . . . . . . . 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 2517 . . . . . . . . 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 3351 . . . . . . . 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 19767 . . . . . . . . . 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 621 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  ->  A. x  e.  a  E. u  e.  U  ( u " { x } ) 
C_  a )
5352r19.21bi 2812 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  /\  x  e.  a )  ->  E. u  e.  U  ( u " { x } ) 
C_  a )
5416, 45, 50, 53syl21anc 1212 . . . . . . 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 2860 . . . . . 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 462 . . . . . 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 2860 . . . . 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 2797 . . . 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 19763 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
60 elutop 19767 . . . . . 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 482 . . . 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 1211 . . 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 3359 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 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3324    C_ wss 3325   {csn 3874    X. cxp 4834   ran crn 4837   "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:  restutopopn  19772
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