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Theorem utopval 20713
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 20712 . . 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 4244 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5195 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
6 ustbas2 20706 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
76adantr 465 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  X  =  dom  U. U )
85, 7eqtr4d 2487 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  X )
98pweqd 4002 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ~P dom  U. u  =  ~P X )
103rexeqdv 3047 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ( E. v  e.  u  ( v " {
x } )  C_  a 
<->  E. v  e.  U  ( v " {
x } )  C_  a ) )
1110ralbidv 2882 . . 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 3090 . 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 20705 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
14 elfvex 5883 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
15 pwexg 4621 . . 3  |-  ( X  e.  _V  ->  ~P X  e.  _V )
16 rabexg 4587 . . 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 5946 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 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   {crab 2797   _Vcvv 3095    C_ wss 3461   ~Pcpw 3997   {csn 4014   U.cuni 4234    |-> cmpt 4495   dom cdm 4989   ran crn 4990   "cima 4992   ` cfv 5578  UnifOncust 20680  unifTopcutop 20711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586  df-ust 20681  df-utop 20712
This theorem is referenced by:  elutop  20714  utoptop  20715  utopbas  20716  utopsnneiplem  20728  metutopOLD  21063  psmetutop  21064
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