MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustbas2 Structured version   Unicode version

Theorem ustbas2 19927
Description: Second direction for ustbas 19929. (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 5162 . 2  |-  dom  ( X  X.  X )  =  X
2 ustbasel 19908 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
3 elssuni 4224 . . . . 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 5821 . . . . . . . . 9  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
6 isust 19905 . . . . . . . . 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 1000 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  U  C_  ~P ( X  X.  X
) )
109unissd 4218 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  U. ~P ( X  X.  X ) )
11 unipw 4645 . . . . 5  |-  U. ~P ( X  X.  X
)  =  ( X  X.  X )
1210, 11syl6sseq 3505 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  U. U  C_  ( X  X.  X
) )
134, 12eqssd 3476 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  =  U. U )
1413dmeqd 5145 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  dom  ( X  X.  X )  =  dom  U. U )
151, 14syl5eqr 2507 1  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797   _Vcvv 3072    i^i cin 3430    C_ wss 3431   ~Pcpw 3963   U.cuni 4194    _I cid 4734    X. cxp 4941   `'ccnv 4942   dom cdm 4943    |` cres 4945    o. ccom 4947   ` cfv 5521  UnifOncust 19901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-res 4955  df-iota 5484  df-fun 5523  df-fv 5529  df-ust 19902
This theorem is referenced by:  ustbas  19929  utopval  19934  tuslem  19969  ucnval  19979  iscfilu  19990
  Copyright terms: Public domain W3C validator