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Theorem mreunirn 15459
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 15449 . . . 4  |- Moore  Fn  _V
2 fnunirn 6164 . . . 4  |-  (Moore  Fn  _V  ->  ( C  e. 
U. ran Moore  <->  E. x  e.  _V  C  e.  (Moore `  x
) ) )
31, 2ax-mp 5 . . 3  |-  ( C  e.  U. ran Moore  <->  E. x  e.  _V  C  e.  (Moore `  x ) )
4 mreuni 15458 . . . . . . 7  |-  ( C  e.  (Moore `  x
)  ->  U. C  =  x )
54fveq2d 5876 . . . . . 6  |-  ( C  e.  (Moore `  x
)  ->  (Moore `  U. C )  =  (Moore `  x ) )
65eleq2d 2490 . . . . 5  |-  ( C  e.  (Moore `  x
)  ->  ( C  e.  (Moore `  U. C )  <-> 
C  e.  (Moore `  x ) ) )
76ibir 245 . . . 4  |-  ( C  e.  (Moore `  x
)  ->  C  e.  (Moore `  U. C ) )
87rexlimivw 2912 . . 3  |-  ( E. x  e.  _V  C  e.  (Moore `  x )  ->  C  e.  (Moore `  U. C ) )
93, 8sylbi 198 . 2  |-  ( C  e.  U. ran Moore  ->  C  e.  (Moore `  U. C ) )
10 fvssunirn 5895 . . 3  |-  (Moore `  U. C )  C_  U. ran Moore
1110sseli 3457 . 2  |-  ( C  e.  (Moore `  U. C )  ->  C  e.  U. ran Moore )
129, 11impbii 190 1  |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1867   E.wrex 2774   _Vcvv 3078   U.cuni 4213   ran crn 4846    Fn wfn 5587   ` cfv 5592  Moorecmre 15440
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
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 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-mre 15444
This theorem is referenced by:  fnmrc  15465  mrcfval  15466
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