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Theorem ustfn 20434
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 4012 . . . . 5  |-  ( u  e.  ~P ~P (
x  X.  x )  <-> 
u  C_  ~P (
x  X.  x ) )
21abbii 2596 . . . 4  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  {
u  |  u  C_  ~P ( x  X.  x
) }
3 abid2 2602 . . . . 5  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  =  ~P ~P ( x  X.  x
)
4 vex 3111 . . . . . . . 8  |-  x  e. 
_V
54, 4xpex 6706 . . . . . . 7  |-  ( x  X.  x )  e. 
_V
65pwex 4625 . . . . . 6  |-  ~P (
x  X.  x )  e.  _V
76pwex 4625 . . . . 5  |-  ~P ~P ( x  X.  x
)  e.  _V
83, 7eqeltri 2546 . . . 4  |-  { u  |  u  e.  ~P ~P ( x  X.  x
) }  e.  _V
92, 8eqeltrri 2547 . . 3  |-  { u  |  u  C_  ~P (
x  X.  x ) }  e.  _V
10 simp1 991 . . . 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 3567 . . 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 4587 . 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 20433 . 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 5702 1  |- UnifOn  Fn  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   _Vcvv 3108    i^i cin 3470    C_ wss 3471   ~Pcpw 4005    _I cid 4785    X. cxp 4992   `'ccnv 4993    |` cres 4996    o. ccom 4998    Fn wfn 5576  UnifOncust 20432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-fun 5583  df-fn 5584  df-ust 20433
This theorem is referenced by:  ustn0  20453  elrnust  20457  ustbas  20460
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