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Theorem elrnust 21170
Description: First direction for ustbas 21173. (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 5907 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  dom UnifOn )
2 fveq2 5881 . . . . 5  |-  ( x  =  X  ->  (UnifOn `  x )  =  (UnifOn `  X ) )
32eleq2d 2499 . . . 4  |-  ( x  =  X  ->  ( U  e.  (UnifOn `  x
)  <->  U  e.  (UnifOn `  X ) ) )
43rspcev 3188 . . 3  |-  ( ( X  e.  dom UnifOn  /\  U  e.  (UnifOn `  X )
)  ->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x ) )
51, 4mpancom 673 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  E. x  e.  dom UnifOn U  e.  (UnifOn `  x ) )
6 ustfn 21147 . . 3  |- UnifOn  Fn  _V
7 fnfun 5691 . . 3  |-  (UnifOn  Fn  _V  ->  Fun UnifOn )
8 elunirn 6171 . . 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 215 1  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870   E.wrex 2783   _Vcvv 3087   U.cuni 4222   dom cdm 4854   ran crn 4855   Fun wfun 5595    Fn wfn 5596   ` cfv 5601  UnifOncust 21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ust 21146
This theorem is referenced by:  ustbas  21173  utopval  21178  tusval  21212  ucnval  21223  iscfilu  21234
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