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Theorem utopval 18254
Description: The topology induced by a uniform structure  U. (Contributed by Thierry Arnoux, 30-Nov-2017.)
Assertion
Ref Expression
utopval  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a } )
Distinct variable groups:    v, a, x, U    X, a, x
Allowed substitution hint:    X( v)

Proof of Theorem utopval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 df-utop 18253 . . 3  |- unifTop  =  ( u  e.  U. ran UnifOn  |->  { a  e.  ~P dom  U. u  |  A. x  e.  a  E. v  e.  u  ( v " { x } ) 
C_  a } )
21a1i 11 . 2  |-  ( U  e.  (UnifOn `  X
)  -> unifTop  =  ( u  e.  U. ran UnifOn  |->  { a  e.  ~P dom  U. u  |  A. x  e.  a  E. v  e.  u  ( v " { x } ) 
C_  a } ) )
3 simpr 448 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  u  =  U )
43unieqd 4018 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5064 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
6 ustbas2 18247 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
76adantr 452 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  X  =  dom  U. U )
85, 7eqtr4d 2470 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  X )
98pweqd 3796 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ~P dom  U. u  =  ~P X )
103rexeqdv 2903 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ( E. v  e.  u  ( v " {
x } )  C_  a 
<->  E. v  e.  U  ( v " {
x } )  C_  a ) )
1110ralbidv 2717 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ( A. x  e.  a  E. v  e.  u  ( v " {
x } )  C_  a 
<-> 
A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a ) )
129, 11rabeqbidv 2943 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  { a  e.  ~P dom  U. u  |  A. x  e.  a  E. v  e.  u  ( v " { x } ) 
C_  a }  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " { x } ) 
C_  a } )
13 elrnust 18246 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
14 elfvex 5750 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
15 pwexg 4375 . . 3  |-  ( X  e.  _V  ->  ~P X  e.  _V )
16 rabexg 4345 . . 3  |-  ( ~P X  e.  _V  ->  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a }  e.  _V )
1714, 15, 163syl 19 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  (
v " { x } )  C_  a }  e.  _V )
182, 12, 13, 17fvmptd 5802 1  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007    e. cmpt 4258   dom cdm 4870   ran crn 4871   "cima 4873   ` cfv 5446  UnifOncust 18221  unifTopcutop 18252
This theorem is referenced by:  elutop  18255  utoptop  18256  utopbas  18257  utopsnneiplem  18269  metutopOLD  18604  psmetutop  18605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ust 18222  df-utop 18253
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