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Theorem elutop 19935
Description: Open sets in the topology induced by an uniform structure  U on  X (Contributed by Thierry Arnoux, 30-Nov-2017.)
Assertion
Ref Expression
elutop  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  (unifTop `  U )  <->  ( A  C_  X  /\  A. x  e.  A  E. v  e.  U  (
v " { x } )  C_  A
) ) )
Distinct variable groups:    x, v, A    v, U, x    x, X
Allowed substitution hint:    X( v)

Proof of Theorem elutop
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 utopval 19934 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a } )
21eleq2d 2522 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  (unifTop `  U )  <->  A  e.  { a  e. 
~P X  |  A. x  e.  a  E. v  e.  U  (
v " { x } )  C_  a } ) )
3 sseq2 3481 . . . . . 6  |-  ( a  =  A  ->  (
( v " {
x } )  C_  a 
<->  ( v " {
x } )  C_  A ) )
43rexbidv 2857 . . . . 5  |-  ( a  =  A  ->  ( E. v  e.  U  ( v " {
x } )  C_  a 
<->  E. v  e.  U  ( v " {
x } )  C_  A ) )
54raleqbi1dv 3025 . . . 4  |-  ( a  =  A  ->  ( A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a 
<-> 
A. x  e.  A  E. v  e.  U  ( v " {
x } )  C_  A ) )
65elrab 3218 . . 3  |-  ( A  e.  { a  e. 
~P X  |  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 ) )
72, 6syl6bb 261 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  (unifTop `  U )  <->  ( A  e.  ~P X  /\  A. x  e.  A  E. v  e.  U  ( v " {
x } )  C_  A ) ) )
8 elex 3081 . . . . 5  |-  ( A  e.  ~P X  ->  A  e.  _V )
98a1i 11 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  ~P X  ->  A  e.  _V ) )
10 elfvex 5821 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
1110adantr 465 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
12 simpr 461 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
1311, 12ssexd 4542 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
1413ex 434 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  C_  X  ->  A  e.  _V ) )
15 elpwg 3971 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
1615a1i 11 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  _V  ->  ( A  e.  ~P X  <->  A  C_  X
) ) )
179, 14, 16pm5.21ndd 354 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
1817anbi1d 704 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( ( A  e.  ~P X  /\  A. x  e.  A  E. v  e.  U  ( v " {
x } )  C_  A )  <->  ( A  C_  X  /\  A. x  e.  A  E. v  e.  U  ( v " { x } ) 
C_  A ) ) )
197, 18bitrd 253 1  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  (unifTop `  U )  <->  ( A  C_  X  /\  A. x  e.  A  E. v  e.  U  (
v " { x } )  C_  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797   {crab 2800   _Vcvv 3072    C_ wss 3431   ~Pcpw 3963   {csn 3980   "cima 4946   ` cfv 5521  UnifOncust 19901  unifTopcutop 19932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529  df-ust 19902  df-utop 19933
This theorem is referenced by:  utoptop  19936  utopbas  19937  restutop  19939  restutopopn  19940  ucncn  19987
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