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Theorem restutop 20906
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 455 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
( U  e.  (UnifOn `  X )  /\  A  C_  X ) )
2 fvex 5858 . . . . . . . 8  |-  (unifTop `  U
)  e.  _V
32a1i 11 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (unifTop `  U )  e.  _V )
4 elfvex 5875 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
54adantr 463 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
6 simpr 459 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
75, 6ssexd 4584 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
8 elrest 14917 . . . . . . 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 659 . . . . . 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 482 . . . . 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 3705 . . . . . . 7  |-  ( a  i^i  A )  C_  A
12 sseq1 3510 . . . . . . 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 2943 . . . . 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 723 . . . . . . . . 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 733 . . . . . . . . . 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 6575 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2018, 18, 19syl2anc 659 . . . . . . . . 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 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
)  ->  u  e.  U )
22 elrestr 14918 . . . . . . . . 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 1226 . . . . . . . 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 3704 . . . . . . . . . . . . 13  |-  ( u  i^i  ( A  X.  A ) )  C_  u
25 imass1 5359 . . . . . . . . . . . . 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 3497 . . . . . . . . . . . 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 668 . . . . . . . . . . 11  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  a )
29 imassrn 5336 . . . . . . . . . . . . . . 15  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( u  i^i  ( A  X.  A ) )
30 rnin 5400 . . . . . . . . . . . . . . 15  |-  ran  (
u  i^i  ( A  X.  A ) )  C_  ( ran  u  i^i  ran  ( A  X.  A
) )
3129, 30sstri 3498 . . . . . . . . . . . . . 14  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( ran  u  i^i  ran  ( A  X.  A ) )
32 inss2 3705 . . . . . . . . . . . . . 14  |-  ( ran  u  i^i  ran  ( A  X.  A ) ) 
C_  ran  ( A  X.  A )
3331, 32sstri 3498 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( A  X.  A )
34 rnxpid 5425 . . . . . . . . . . . . 13  |-  ran  ( A  X.  A )  =  A
3533, 34sseqtri 3521 . . . . . . . . . . . 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 3708 . . . . . . . . . 10  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  ( a  i^i  A
) )
3837adantl 464 . . . . . . . . 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 3526 . . . . . . . 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 5320 . . . . . . . . . 10  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
v " { x } )  =  ( ( u  i^i  ( A  X.  A ) )
" { x }
) )
4241sseq1d 3516 . . . . . . . . 9  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
( v " {
x } )  C_  b 
<->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  b ) )
4342rspcev 3207 . . . . . . . 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 659 . . . . . . 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 753 . . . . . . . 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 3704 . . . . . . . . 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 459 . . . . . . . . . 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 2544 . . . . . . . . 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 3487 . . . . . . . 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 20902 . . . . . . . . . 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 622 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  ->  A. x  e.  a  E. u  e.  U  ( u " { x } ) 
C_  a )
5352r19.21bi 2823 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  /\  x  e.  a )  ->  E. u  e.  U  ( u " { x } ) 
C_  a )
5416, 45, 50, 53syl21anc 1225 . . . . . . 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 2996 . . . . . 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 463 . . . . . 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 2996 . . . . 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 2868 . . . 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 20898 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
60 elutop 20902 . . . . . 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 483 . . . 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 1224 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )
6463ex 432 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
b  e.  ( (unifTop `  U )t  A )  ->  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) ) )
6564ssrdv 3495 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   {csn 4016    X. cxp 4986   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   ↾t crest 14910  UnifOncust 20868  unifTopcutop 20899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-rest 14912  df-ust 20869  df-utop 20900
This theorem is referenced by:  restutopopn  20907
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