MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elutop Structured version   Unicode version

Theorem elutop 20902
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 20901 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " {
x } )  C_  a } )
21eleq2d 2524 . . 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 3511 . . . . . 6  |-  ( a  =  A  ->  (
( v " {
x } )  C_  a 
<->  ( v " {
x } )  C_  A ) )
43rexbidv 2965 . . . . 5  |-  ( a  =  A  ->  ( E. v  e.  U  ( v " {
x } )  C_  a 
<->  E. v  e.  U  ( v " {
x } )  C_  A ) )
54raleqbi1dv 3059 . . . 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 3254 . . 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 3115 . . . . 5  |-  ( A  e.  ~P X  ->  A  e.  _V )
98a1i 11 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  ~P X  ->  A  e.  _V ) )
10 elfvex 5875 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
1110adantr 463 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  X  e.  _V )
12 simpr 459 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  C_  X )
1311, 12ssexd 4584 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
1413ex 432 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  C_  X  ->  A  e.  _V ) )
15 elpwg 4007 . . . . 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 352 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
1817anbi1d 702 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   {csn 4016   "cima 4991   ` cfv 5570  UnifOncust 20868  unifTopcutop 20899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-ust 20869  df-utop 20900
This theorem is referenced by:  utoptop  20903  utopbas  20904  restutop  20906  restutopopn  20907  ucncn  20954
  Copyright terms: Public domain W3C validator