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Theorem utopbas 20470
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 20467 . . . 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 3585 . . . 4  |-  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a }  C_  ~P X
31, 2syl6eqss 3554 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  C_  ~P X
)
4 ssid 3523 . . . . . 6  |-  X  C_  X
54a1i 11 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  C_  X
)
6 ustssxp 20439 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  v  C_  ( X  X.  X
) )
7 imassrn 5346 . . . . . . . . . 10  |-  ( v
" { x }
)  C_  ran  v
8 rnss 5229 . . . . . . . . . . 11  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  ran  ( X  X.  X ) )
9 rnxpid 5438 . . . . . . . . . . 11  |-  ran  ( X  X.  X )  =  X
108, 9syl6sseq 3550 . . . . . . . . . 10  |-  ( v 
C_  ( X  X.  X )  ->  ran  v  C_  X )
117, 10syl5ss 3515 . . . . . . . . 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 2878 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  A. v  e.  U  ( v " { x } ) 
C_  X )
14 ustne0 20448 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
15 r19.2zb 3918 . . . . . . . 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 2879 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  A. x  e.  X  E. v  e.  U  ( v " { x } ) 
C_  X )
19 elutop 20468 . . . . 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 920 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  (unifTop `  U ) )
21 elpwuni 4413 . . . 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 2475 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245    X. cxp 4997   ran crn 5000   "cima 5002   ` cfv 5586  UnifOncust 20434  unifTopcutop 20465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ust 20435  df-utop 20466
This theorem is referenced by:  utoptopon  20471  utop2nei  20485  utopreg  20487  tuslem  20502
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