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Theorem utopval 19806
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 19805 . . 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 4100 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  U. u  =  U. U )
54dmeqd 5041 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  dom  U. U )
6 ustbas2 19799 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
76adantr 465 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  X  =  dom  U. U )
85, 7eqtr4d 2477 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  dom  U. u  =  X )
98pweqd 3864 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ~P dom  U. u  =  ~P X )
103rexeqdv 2923 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  u  =  U )  ->  ( E. v  e.  u  ( v " {
x } )  C_  a 
<->  E. v  e.  U  ( v " {
x } )  C_  a ) )
1110ralbidv 2734 . . 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 2966 . 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 19798 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
14 elfvex 5716 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
15 pwexg 4475 . . 3  |-  ( X  e.  _V  ->  ~P X  e.  _V )
16 rabexg 4441 . . 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 5778 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 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 2971    C_ wss 3327   ~Pcpw 3859   {csn 3876   U.cuni 4090    e. cmpt 4349   dom cdm 4839   ran crn 4840   "cima 4842   ` cfv 5417  UnifOncust 19773  unifTopcutop 19804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5380  df-fun 5419  df-fn 5420  df-fv 5425  df-ust 19774  df-utop 19805
This theorem is referenced by:  elutop  19807  utoptop  19808  utopbas  19809  utopsnneiplem  19821  metutopOLD  20156  psmetutop  20157
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