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Theorem utopbas 20863
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 20860 . . . 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 3581 . . . 4  |-  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a }  C_  ~P X
31, 2syl6eqss 3549 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  C_  ~P X
)
4 ssid 3518 . . . . . 6  |-  X  C_  X
54a1i 11 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  C_  X
)
6 ustssxp 20832 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  v  C_  ( X  X.  X
) )
7 imassrn 5358 . . . . . . . . . 10  |-  ( v
" { x }
)  C_  ran  v
8 rnss 5241 . . . . . . . . . . 11  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  ran  ( X  X.  X ) )
9 rnxpid 5447 . . . . . . . . . . 11  |-  ran  ( X  X.  X )  =  X
108, 9syl6sseq 3545 . . . . . . . . . 10  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  X )
117, 10syl5ss 3510 . . . . . . . . 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 2871 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  A. v  e.  U  ( v " { x } ) 
C_  X )
14 ustne0 20841 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
15 r19.2zb 3922 . . . . . . . 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 2872 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  A. x  e.  X  E. v  e.  U  ( v " { x } ) 
C_  X )
19 elutop 20861 . . . . 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 922 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  (unifTop `  U ) )
21 elpwuni 4423 . . . 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 2465 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U.cuni 4251    X. cxp 5006   ran crn 5009   "cima 5011   ` cfv 5594  UnifOncust 20827  unifTopcutop 20858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ust 20828  df-utop 20859
This theorem is referenced by:  utoptopon  20864  utop2nei  20878  utopreg  20880  tuslem  20895
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