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Theorem mrcidb 14887
Description: A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcidb  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )

Proof of Theorem mrcidb
StepHypRef Expression
1 mrcfval.f . . 3  |-  F  =  (mrCls `  C )
21mrcid 14885 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )
3 simpr 461 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  ( F `  U )  =  U )
41mrcssv 14886 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U )  C_  X
)
54adantr 465 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  ( F `  U )  C_  X )
63, 5eqsstr3d 3544 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  U  C_  X )
71mrccl 14883 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
86, 7syldan 470 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  ( F `  U )  e.  C )
93, 8eqeltrrd 2556 . 2  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  U  e.  C )
102, 9impbida 830 1  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   ` cfv 5594  Moorecmre 14854  mrClscmrc 14855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 14858  df-mrc 14859
This theorem is referenced by:  mrcidb2  14890
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