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Theorem ustbas 20912
Description: Recover the base of an uniform structure  U.  U. ran UnifOn is to UnifOn what  Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypothesis
Ref Expression
ustbas.1  |-  X  =  dom  U. U
Assertion
Ref Expression
ustbas  |-  ( U  e.  U. ran UnifOn  <->  U  e.  (UnifOn `  X ) )

Proof of Theorem ustbas
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ustfn 20886 . . . 4  |- UnifOn  Fn  _V
2 fnfun 5613 . . . 4  |-  (UnifOn  Fn  _V  ->  Fun UnifOn )
3 elunirn 6098 . . . 4  |-  ( Fun UnifOn  ->  ( U  e.  U. ran UnifOn  <->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x
) ) )
41, 2, 3mp2b 10 . . 3  |-  ( U  e.  U. ran UnifOn  <->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x ) )
5 ustbas2 20910 . . . . . . . 8  |-  ( U  e.  (UnifOn `  x
)  ->  x  =  dom  U. U )
6 ustbas.1 . . . . . . . 8  |-  X  =  dom  U. U
75, 6syl6eqr 2459 . . . . . . 7  |-  ( U  e.  (UnifOn `  x
)  ->  x  =  X )
87fveq2d 5807 . . . . . 6  |-  ( U  e.  (UnifOn `  x
)  ->  (UnifOn `  x
)  =  (UnifOn `  X ) )
98eleq2d 2470 . . . . 5  |-  ( U  e.  (UnifOn `  x
)  ->  ( U  e.  (UnifOn `  x )  <->  U  e.  (UnifOn `  X
) ) )
109ibi 241 . . . 4  |-  ( U  e.  (UnifOn `  x
)  ->  U  e.  (UnifOn `  X ) )
1110rexlimivw 2890 . . 3  |-  ( E. x  e.  dom UnifOn U  e.  (UnifOn `  x )  ->  U  e.  (UnifOn `  X ) )
124, 11sylbi 195 . 2  |-  ( U  e.  U. ran UnifOn  ->  U  e.  (UnifOn `  X )
)
13 elrnust 20909 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
1412, 13impbii 188 1  |-  ( U  e.  U. ran UnifOn  <->  U  e.  (UnifOn `  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1403    e. wcel 1840   E.wrex 2752   _Vcvv 3056   U.cuni 4188   dom cdm 4940   ran crn 4941   Fun wfun 5517    Fn wfn 5518   ` cfv 5523  UnifOncust 20884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531  df-ust 20885
This theorem is referenced by: (None)
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