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Theorem mreunirn 13789
Description: Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mreunirn  |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )

Proof of Theorem mreunirn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fnmre 13779 . . . 4  |- Moore  Fn  _V
2 fnunirn 5966 . . . 4  |-  (Moore  Fn  _V  ->  ( C  e. 
U. ran Moore  <->  E. x  e.  _V  C  e.  (Moore `  x
) ) )
31, 2ax-mp 8 . . 3  |-  ( C  e.  U. ran Moore  <->  E. x  e.  _V  C  e.  (Moore `  x ) )
4 mreuni 13788 . . . . . . 7  |-  ( C  e.  (Moore `  x
)  ->  U. C  =  x )
54fveq2d 5699 . . . . . 6  |-  ( C  e.  (Moore `  x
)  ->  (Moore `  U. C )  =  (Moore `  x ) )
65eleq2d 2479 . . . . 5  |-  ( C  e.  (Moore `  x
)  ->  ( C  e.  (Moore `  U. C )  <-> 
C  e.  (Moore `  x ) ) )
76ibir 234 . . . 4  |-  ( C  e.  (Moore `  x
)  ->  C  e.  (Moore `  U. C ) )
87rexlimivw 2794 . . 3  |-  ( E. x  e.  _V  C  e.  (Moore `  x )  ->  C  e.  (Moore `  U. C ) )
93, 8sylbi 188 . 2  |-  ( C  e.  U. ran Moore  ->  C  e.  (Moore `  U. C ) )
10 fvssunirn 5721 . . 3  |-  (Moore `  U. C )  C_  U. ran Moore
1110sseli 3312 . 2  |-  ( C  e.  (Moore `  U. C )  ->  C  e.  U. ran Moore )
129, 11impbii 181 1  |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1721   E.wrex 2675   _Vcvv 2924   U.cuni 3983   ran crn 4846    Fn wfn 5416   ` cfv 5421  Moorecmre 13770
This theorem is referenced by:  fnmrc  13795  mrcfval  13796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-mre 13774
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