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Theorem mreunirn 13596
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 13586 . . . 4  |- Moore  Fn  _V
2 fnunirn 5862 . . . 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 13595 . . . . . . 7  |-  ( C  e.  (Moore `  x
)  ->  U. C  =  x )
54fveq2d 5609 . . . . . 6  |-  ( C  e.  (Moore `  x
)  ->  (Moore `  U. C )  =  (Moore `  x ) )
65eleq2d 2425 . . . . 5  |-  ( C  e.  (Moore `  x
)  ->  ( C  e.  (Moore `  U. C )  <-> 
C  e.  (Moore `  x ) ) )
76ibir 233 . . . 4  |-  ( C  e.  (Moore `  x
)  ->  C  e.  (Moore `  U. C ) )
87rexlimivw 2739 . . 3  |-  ( E. x  e.  _V  C  e.  (Moore `  x )  ->  C  e.  (Moore `  U. C ) )
93, 8sylbi 187 . 2  |-  ( C  e.  U. ran Moore  ->  C  e.  (Moore `  U. C ) )
10 fvssunirn 5631 . . 3  |-  (Moore `  U. C )  C_  U. ran Moore
1110sseli 3252 . 2  |-  ( C  e.  (Moore `  U. C )  ->  C  e.  U. ran Moore )
129, 11impbii 180 1  |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1710   E.wrex 2620   _Vcvv 2864   U.cuni 3906   ran crn 4769    Fn wfn 5329   ` cfv 5334  Moorecmre 13577
This theorem is referenced by:  fnmrc  13602  mrcfval  13603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-fv 5342  df-mre 13581
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