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Theorem ustbas2 20463
Description: Second direction for ustbas 20465. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ustbas2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )

Proof of Theorem ustbas2
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmxpid 5220 . 2  |-  dom  ( X  X.  X )  =  X
2 ustbasel 20444 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
3 elssuni 4275 . . . . 5  |-  ( ( X  X.  X )  e.  U  ->  ( X  X.  X )  C_  U. U )
42, 3syl 16 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  C_  U. U
)
5 elfvex 5891 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
6 isust 20441 . . . . . . . . 9  |-  ( X  e.  _V  ->  ( U  e.  (UnifOn `  X
)  <->  ( 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 ) ) ) ) )
75, 6syl 16 . . . . . . . 8  |-  ( U  e.  (UnifOn `  X
)  ->  ( U  e.  (UnifOn `  X )  <->  ( 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 ) ) ) ) )
87ibi 241 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  ( 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 ) ) ) )
98simp1d 1008 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U  C_  ~P ( X  X.  X
) )
109unissd 4269 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  U. ~P ( X  X.  X ) )
11 unipw 4697 . . . . 5  |-  U. ~P ( X  X.  X
)  =  ( X  X.  X )
1210, 11syl6sseq 3550 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  ( X  X.  X
) )
134, 12eqssd 3521 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. U )
1413dmeqd 5203 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  dom  ( X  X.  X )  =  dom  U. U )
151, 14syl5eqr 2522 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999    |` cres 5001    o. ccom 5003   ` cfv 5586  UnifOncust 20437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594  df-ust 20438
This theorem is referenced by:  ustbas  20465  utopval  20470  tuslem  20505  ucnval  20515  iscfilu  20526
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