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Theorem ustbas 20460
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 20434 . . . 4  |- UnifOn  Fn  _V
2 fnfun 5671 . . . 4  |-  (UnifOn  Fn  _V  ->  Fun UnifOn )
3 elunirn 6144 . . . 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 20458 . . . . . . . 8  |-  ( U  e.  (UnifOn `  x
)  ->  x  =  dom  U. U )
6 ustbas.1 . . . . . . . 8  |-  X  =  dom  U. U
75, 6syl6eqr 2521 . . . . . . 7  |-  ( U  e.  (UnifOn `  x
)  ->  x  =  X )
87fveq2d 5863 . . . . . 6  |-  ( U  e.  (UnifOn `  x
)  ->  (UnifOn `  x
)  =  (UnifOn `  X ) )
98eleq2d 2532 . . . . 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 2947 . . 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 20457 . 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 1374    e. wcel 1762   E.wrex 2810   _Vcvv 3108   U.cuni 4240   dom cdm 4994   ran crn 4995   Fun wfun 5575    Fn wfn 5576   ` cfv 5581  UnifOncust 20432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589  df-ust 20433
This theorem is referenced by: (None)
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