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Theorem elrnust 19930
Description: First direction for ustbas 19933. (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 5824 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  dom UnifOn )
2 fveq2 5798 . . . . 5  |-  ( x  =  X  ->  (UnifOn `  x )  =  (UnifOn `  X ) )
32eleq2d 2524 . . . 4  |-  ( x  =  X  ->  ( U  e.  (UnifOn `  x
)  <->  U  e.  (UnifOn `  X ) ) )
43rspcev 3177 . . 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 19907 . . 3  |- UnifOn  Fn  _V
7 fnfun 5615 . . 3  |-  (UnifOn  Fn  _V  ->  Fun UnifOn )
8 elunirn 6076 . . 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 1370    e. wcel 1758   E.wrex 2799   _Vcvv 3076   U.cuni 4198   dom cdm 4947   ran crn 4948   Fun wfun 5519    Fn wfn 5520   ` cfv 5525  UnifOncust 19905
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 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-fv 5533  df-ust 19906
This theorem is referenced by:  ustbas  19933  utopval  19938  tusval  19972  ucnval  19983  iscfilu  19994
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