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Theorem elrnust 20490
Description: First direction for ustbas 20493. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
elrnust  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )

Proof of Theorem elrnust
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5892 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  dom UnifOn )
2 fveq2 5866 . . . . 5  |-  ( x  =  X  ->  (UnifOn `  x )  =  (UnifOn `  X ) )
32eleq2d 2537 . . . 4  |-  ( x  =  X  ->  ( U  e.  (UnifOn `  x
)  <->  U  e.  (UnifOn `  X ) ) )
43rspcev 3214 . . 3  |-  ( ( X  e.  dom UnifOn  /\  U  e.  (UnifOn `  X )
)  ->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x ) )
51, 4mpancom 669 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x ) )
6 ustfn 20467 . . 3  |- UnifOn  Fn  _V
7 fnfun 5678 . . 3  |-  (UnifOn  Fn  _V  ->  Fun UnifOn )
8 elunirn 6151 . . 3  |-  ( Fun UnifOn  ->  ( U  e.  U. ran UnifOn  <->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x
) ) )
96, 7, 8mp2b 10 . 2  |-  ( U  e.  U. ran UnifOn  <->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x ) )
105, 9sylibr 212 1  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   U.cuni 4245   dom cdm 4999   ran crn 5000   Fun wfun 5582    Fn wfn 5583   ` cfv 5588  UnifOncust 20465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-ust 20466
This theorem is referenced by:  ustbas  20493  utopval  20498  tusval  20532  ucnval  20543  iscfilu  20554
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