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Theorem elrnust 19774
Description: First direction for ustbas 19777. (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 5711 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  dom UnifOn )
2 fveq2 5686 . . . . 5  |-  ( x  =  X  ->  (UnifOn `  x )  =  (UnifOn `  X ) )
32eleq2d 2505 . . . 4  |-  ( x  =  X  ->  ( U  e.  (UnifOn `  x
)  <->  U  e.  (UnifOn `  X ) ) )
43rspcev 3068 . . 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 19751 . . 3  |- UnifOn  Fn  _V
7 fnfun 5503 . . 3  |-  (UnifOn  Fn  _V  ->  Fun UnifOn )
8 elunirn 5963 . . 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 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967   U.cuni 4086   dom cdm 4835   ran crn 4836   Fun wfun 5407    Fn wfn 5408   ` cfv 5413  UnifOncust 19749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-ust 19750
This theorem is referenced by:  ustbas  19777  utopval  19782  tusval  19816  ucnval  19827  iscfilu  19838
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