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Theorem utopval 20465
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 20464 . . 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 461 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  u  =  U )
43unieqd 4250 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5198 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
6 ustbas2 20458 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
76adantr 465 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  X  =  dom  U. U )
85, 7eqtr4d 2506 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  X )
98pweqd 4010 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ~P dom  U. u  =  ~P X )
103rexeqdv 3060 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ( E. v  e.  u  ( v " {
x } )  C_  a 
<->  E. v  e.  U  ( v " {
x } )  C_  a ) )
1110ralbidv 2898 . . 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 3103 . 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 20457 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
14 elfvex 5886 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
15 pwexg 4626 . . 3  |-  ( X  e.  _V  ->  ~P X  e.  _V )
16 rabexg 4592 . . 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 20 . 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 5948 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   {crab 2813   _Vcvv 3108    C_ wss 3471   ~Pcpw 4005   {csn 4022   U.cuni 4240    |-> cmpt 4500   dom cdm 4994   ran crn 4995   "cima 4997   ` cfv 5581  UnifOncust 20432  unifTopcutop 20463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589  df-ust 20433  df-utop 20464
This theorem is referenced by:  elutop  20466  utoptop  20467  utopbas  20468  utopsnneiplem  20480  metutopOLD  20815  psmetutop  20816
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