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Theorem ustfn 20998
Description: The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ustfn  |- UnifOn  Fn  _V

Proof of Theorem ustfn
Dummy variables  v  u  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 selpw 3964 . . . . 5  |-  ( u  e.  ~P ~P (
x  X.  x )  <-> 
u  C_  ~P (
x  X.  x ) )
21abbii 2538 . . . 4  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  {
u  |  u  C_  ~P ( x  X.  x
) }
3 abid2 2544 . . . . 5  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  ~P ~P ( x  X.  x
)
4 vex 3064 . . . . . . . 8  |-  x  e. 
_V
54, 4xpex 6588 . . . . . . 7  |-  ( x  X.  x )  e. 
_V
65pwex 4579 . . . . . 6  |-  ~P (
x  X.  x )  e.  _V
76pwex 4579 . . . . 5  |-  ~P ~P ( x  X.  x
)  e.  _V
83, 7eqeltri 2488 . . . 4  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  e.  _V
92, 8eqeltrri 2489 . . 3  |-  { u  |  u  C_  ~P (
x  X.  x ) }  e.  _V
10 simp1 999 . . . 4  |-  ( ( u  C_  ~P (
x  X.  x )  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) )  ->  u  C_ 
~P ( x  X.  x ) )
1110ss2abi 3513 . . 3  |-  { u  |  ( u  C_  ~P ( x  X.  x
)  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }  C_  { u  |  u  C_  ~P ( x  X.  x
) }
129, 11ssexi 4541 . 2  |-  { u  |  ( u  C_  ~P ( x  X.  x
)  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) }  e.  _V
13 df-ust 20997 . 2  |- UnifOn  =  ( x  e.  _V  |->  { u  |  ( u 
C_  ~P ( x  X.  x )  /\  (
x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x
) ( v  C_  w  ->  w  e.  u
)  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  ( (  _I  |`  x
)  C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w
)  C_  v )
) ) } )
1412, 13fnmpti 5694 1  |- UnifOn  Fn  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 976    e. wcel 1844   {cab 2389   A.wral 2756   E.wrex 2757   _Vcvv 3061    i^i cin 3415    C_ wss 3416   ~Pcpw 3957    _I cid 4735    X. cxp 4823   `'ccnv 4824    |` cres 4827    o. ccom 4829    Fn wfn 5566  UnifOncust 20996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-fun 5573  df-fn 5574  df-ust 20997
This theorem is referenced by:  ustn0  21017  elrnust  21021  ustbas  21024
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