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Theorem restutop 21262
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 463 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
( U  e.  (UnifOn `  X )  /\  A  C_  X ) )
2 fvex 5857 . . . . . . . 8  |-  (unifTop `  U
)  e.  _V
32a1i 11 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (unifTop `  U )  e.  _V )
4 elfvex 5874 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
54adantr 471 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
6 simpr 467 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
75, 6ssexd 4521 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
8 elrest 15336 . . . . . . 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 671 . . . . . 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 491 . . . . 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 3620 . . . . . . 7  |-  ( a  i^i  A )  C_  A
12 sseq1 3420 . . . . . . 7  |-  ( b  =  ( a  i^i 
A )  ->  (
b  C_  A  <->  ( a  i^i  A )  C_  A
) )
1311, 12mpbiri 241 . . . . . 6  |-  ( b  =  ( a  i^i 
A )  ->  b  C_  A )
1413rexlimivw 2849 . . . . 5  |-  ( E. a  e.  (unifTop `  U
) b  =  ( a  i^i  A )  ->  b  C_  A
)
1510, 14syl 17 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
b  C_  A )
16 simp-5l 783 . . . . . . . . . 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 737 . . . . . . . . 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 747 . . . . . . . . . 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 6580 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2018, 18, 19syl2anc 671 . . . . . . . . 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 767 . . . . . . . . 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 15337 . . . . . . . . 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 1271 . . . . . . . 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 3619 . . . . . . . . . . . . 13  |-  ( u  i^i  ( A  X.  A ) )  C_  u
25 imass1 5180 . . . . . . . . . . . . 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 3407 . . . . . . . . . . . 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 681 . . . . . . . . . . 11  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  a )
29 imassrn 5156 . . . . . . . . . . . . . . 15  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( u  i^i  ( A  X.  A ) )
30 rnin 5222 . . . . . . . . . . . . . . 15  |-  ran  (
u  i^i  ( A  X.  A ) )  C_  ( ran  u  i^i  ran  ( A  X.  A
) )
3129, 30sstri 3408 . . . . . . . . . . . . . 14  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ( ran  u  i^i  ran  ( A  X.  A ) )
32 inss2 3620 . . . . . . . . . . . . . 14  |-  ( ran  u  i^i  ran  ( A  X.  A ) ) 
C_  ran  ( A  X.  A )
3331, 32sstri 3408 . . . . . . . . . . . . 13  |-  ( ( u  i^i  ( A  X.  A ) )
" { x }
)  C_  ran  ( A  X.  A )
34 rnxpid 5247 . . . . . . . . . . . . 13  |-  ran  ( A  X.  A )  =  A
3533, 34sseqtri 3431 . . . . . . . . . . . 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 3623 . . . . . . . . . 10  |-  ( ( u " { x } )  C_  a  ->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  ( a  i^i  A
) )
3837adantl 472 . . . . . . . . 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 774 . . . . . . . . 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 3436 . . . . . . . 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 5140 . . . . . . . . . 10  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
v " { x } )  =  ( ( u  i^i  ( A  X.  A ) )
" { x }
) )
4241sseq1d 3426 . . . . . . . . 9  |-  ( v  =  ( u  i^i  ( A  X.  A
) )  ->  (
( v " {
x } )  C_  b 
<->  ( ( u  i^i  ( A  X.  A
) ) " {
x } )  C_  b ) )
4342rspcev 3117 . . . . . . . 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 671 . . . . . . 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 767 . . . . . . . 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 3619 . . . . . . . . 9  |-  ( a  i^i  A )  C_  a
47 simpllr 774 . . . . . . . . . 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 467 . . . . . . . . . 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 2531 . . . . . . . . 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 3397 . . . . . . . 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 21258 . . . . . . . . . 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 634 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  ->  A. x  e.  a  E. u  e.  U  ( u " { x } ) 
C_  a )
5352r19.21bi 2756 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  a  e.  (unifTop `  U )
)  /\  x  e.  a )  ->  E. u  e.  U  ( u " { x } ) 
C_  a )
5416, 45, 50, 53syl21anc 1270 . . . . . . 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 2899 . . . . . 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 471 . . . . . 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 2899 . . . . 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 2789 . . . 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 21254 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
60 elutop 21258 . . . . . 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 17 . . . . 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 492 . . . 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 1269 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  /\  b  e.  ( (unifTop `  U
)t 
A ) )  -> 
b  e.  (unifTop `  ( Ut  ( A  X.  A
) ) ) )
6463ex 440 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
b  e.  ( (unifTop `  U )t  A )  ->  b  e.  (unifTop `  ( Ut  ( A  X.  A ) ) ) ) )
6564ssrdv 3405 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 189    /\ wa 375    = wceq 1447    e. wcel 1890   A.wral 2736   E.wrex 2737   _Vcvv 3012    i^i cin 3370    C_ wss 3371   {csn 3935    X. cxp 4809   ran crn 4812   "cima 4814   ` cfv 5560  (class class class)co 6275   ↾t crest 15329  UnifOncust 21224  unifTopcutop 21255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6780  df-2nd 6781  df-rest 15331  df-ust 21225  df-utop 21256
This theorem is referenced by:  restutopopn  21263
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