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Theorem utopbas 19943
Description: The base of the topology induced by a uniform structure  U. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
utopbas  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)

Proof of Theorem utopbas
Dummy variables  a 
v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utopval 19940 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a } )
2 ssrab2 3546 . . . 4  |-  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a }  C_  ~P X
31, 2syl6eqss 3515 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  C_  ~P X
)
4 ssid 3484 . . . . . 6  |-  X  C_  X
54a1i 11 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  C_  X
)
6 ustssxp 19912 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  v  C_  ( X  X.  X
) )
7 imassrn 5289 . . . . . . . . . 10  |-  ( v
" { x }
)  C_  ran  v
8 rnss 5177 . . . . . . . . . . 11  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  ran  ( X  X.  X ) )
9 rnxpid 5380 . . . . . . . . . . 11  |-  ran  ( X  X.  X )  =  X
108, 9syl6sseq 3511 . . . . . . . . . 10  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  X )
117, 10syl5ss 3476 . . . . . . . . 9  |-  ( v 
C_  ( X  X.  X )  ->  (
v " { x } )  C_  X
)
126, 11syl 16 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  (
v " { x } )  C_  X
)
1312ralrimiva 2830 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  A. v  e.  U  ( v " { x } ) 
C_  X )
14 ustne0 19921 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
15 r19.2zb 3879 . . . . . . . 8  |-  ( U  =/=  (/)  <->  ( A. v  e.  U  ( v " { x } ) 
C_  X  ->  E. v  e.  U  ( v " { x } ) 
C_  X ) )
1614, 15sylib 196 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  ( A. v  e.  U  (
v " { x } )  C_  X  ->  E. v  e.  U  ( v " {
x } )  C_  X ) )
1713, 16mpd 15 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  E. v  e.  U  ( v " { x } ) 
C_  X )
1817ralrimivw 2831 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  A. x  e.  X  E. v  e.  U  ( v " { x } ) 
C_  X )
19 elutop 19941 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  e.  (unifTop `  U )  <->  ( X  C_  X  /\  A. x  e.  X  E. v  e.  U  (
v " { x } )  C_  X
) ) )
205, 18, 19mpbir2and 913 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  (unifTop `  U ) )
21 elpwuni 4367 . . . 4  |-  ( X  e.  (unifTop `  U
)  ->  ( (unifTop `  U )  C_  ~P X 
<-> 
U. (unifTop `  U )  =  X ) )
2220, 21syl 16 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( (unifTop `  U )  C_  ~P X 
<-> 
U. (unifTop `  U )  =  X ) )
233, 22mpbid 210 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  X )
2423eqcomd 2462 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   {csn 3986   U.cuni 4200    X. cxp 4947   ran crn 4950   "cima 4952   ` cfv 5527  UnifOncust 19907  unifTopcutop 19938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-ust 19908  df-utop 19939
This theorem is referenced by:  utoptopon  19944  utop2nei  19958  utopreg  19960  tuslem  19975
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