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Theorem ustbas2 20912
Description: Second direction for ustbas 20914. (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 5164 . 2  |-  dom  ( X  X.  X )  =  X
2 ustbasel 20893 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
3 elssuni 4219 . . . . 5  |-  ( ( X  X.  X )  e.  U  ->  ( X  X.  X )  C_  U. U )
42, 3syl 17 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  C_  U. U
)
5 elfvex 5832 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
6 isust 20890 . . . . . . . . 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 17 . . . . . . . 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 1009 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U  C_  ~P ( X  X.  X
) )
109unissd 4214 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  U. ~P ( X  X.  X ) )
11 unipw 4640 . . . . 5  |-  U. ~P ( X  X.  X
)  =  ( X  X.  X )
1210, 11syl6sseq 3487 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  ( X  X.  X
) )
134, 12eqssd 3458 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. U )
1413dmeqd 5147 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  dom  ( X  X.  X )  =  dom  U. U )
151, 14syl5eqr 2457 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754   _Vcvv 3058    i^i cin 3412    C_ wss 3413   ~Pcpw 3954   U.cuni 4190    _I cid 4732    X. cxp 4940   `'ccnv 4941   dom cdm 4942    |` cres 4944    o. ccom 4946   ` cfv 5525  UnifOncust 20886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-res 4954  df-iota 5489  df-fun 5527  df-fv 5533  df-ust 20887
This theorem is referenced by:  ustbas  20914  utopval  20919  tuslem  20954  ucnval  20964  iscfilu  20975
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