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Theorem utopbas 21299
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 21296 . . . 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 3526 . . . 4  |-  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a }  C_  ~P X
31, 2syl6eqss 3494 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  C_  ~P X
)
4 ssid 3463 . . . . . 6  |-  X  C_  X
54a1i 11 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  C_  X
)
6 ustssxp 21268 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  v  C_  ( X  X.  X
) )
7 imassrn 5198 . . . . . . . . . 10  |-  ( v
" { x }
)  C_  ran  v
8 rnss 5082 . . . . . . . . . . 11  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  ran  ( X  X.  X ) )
9 rnxpid 5289 . . . . . . . . . . 11  |-  ran  ( X  X.  X )  =  X
108, 9syl6sseq 3490 . . . . . . . . . 10  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  X )
117, 10syl5ss 3455 . . . . . . . . 9  |-  ( v 
C_  ( X  X.  X )  ->  (
v " { x } )  C_  X
)
126, 11syl 17 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  (
v " { x } )  C_  X
)
1312ralrimiva 2814 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  A. v  e.  U  ( v " { x } ) 
C_  X )
14 ustne0 21277 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
15 r19.2zb 3871 . . . . . . . 8  |-  ( U  =/=  (/)  <->  ( A. v  e.  U  ( v " { x } ) 
C_  X  ->  E. v  e.  U  ( v " { x } ) 
C_  X ) )
1614, 15sylib 201 . . . . . . 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 2815 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  A. x  e.  X  E. v  e.  U  ( v " { x } ) 
C_  X )
19 elutop 21297 . . . . 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 938 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  (unifTop `  U ) )
21 elpwuni 4383 . . . 4  |-  ( X  e.  (unifTop `  U
)  ->  ( (unifTop `  U )  C_  ~P X 
<-> 
U. (unifTop `  U )  =  X ) )
2220, 21syl 17 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( (unifTop `  U )  C_  ~P X 
<-> 
U. (unifTop `  U )  =  X ) )
233, 22mpbid 215 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U. (unifTop `  U )  =  X )
2423eqcomd 2468 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   {csn 3980   U.cuni 4212    X. cxp 4851   ran crn 4854   "cima 4856   ` cfv 5601  UnifOncust 21263  unifTopcutop 21294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ust 21264  df-utop 21295
This theorem is referenced by:  utoptopon  21300  utop2nei  21314  utopreg  21316  tuslem  21331
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