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Theorem ustbas2 21217
Description: Second direction for ustbas 21219. (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 5066 . 2  |-  dom  ( X  X.  X )  =  X
2 ustbasel 21198 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
3 elssuni 4242 . . . . 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 5900 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
6 isust 21195 . . . . . . . . 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 244 . . . . . . 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 1017 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U  C_  ~P ( X  X.  X
) )
109unissd 4237 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  U. ~P ( X  X.  X ) )
11 unipw 4664 . . . . 5  |-  U. ~P ( X  X.  X
)  =  ( X  X.  X )
1210, 11syl6sseq 3507 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  ( X  X.  X
) )
134, 12eqssd 3478 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. U )
1413dmeqd 5049 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  dom  ( X  X.  X )  =  dom  U. U )
151, 14syl5eqr 2475 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   _Vcvv 3078    i^i cin 3432    C_ wss 3433   ~Pcpw 3976   U.cuni 4213    _I cid 4756    X. cxp 4844   `'ccnv 4845   dom cdm 4846    |` cres 4848    o. ccom 4850   ` cfv 5593  UnifOncust 21191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-res 4858  df-iota 5557  df-fun 5595  df-fv 5601  df-ust 21192
This theorem is referenced by:  ustbas  21219  utopval  21224  tuslem  21259  ucnval  21269  iscfilu  21280
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