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Theorem mrcidb 14553
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 14551 . 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 14552 . . . . . 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 3391 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  U  C_  X )
71mrccl 14549 . . . 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 2518 . 2  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  =  U )  ->  U  e.  C )
102, 9impbida 828 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 1369    e. wcel 1756    C_ wss 3328   ` cfv 5418  Moorecmre 14520  mrClscmrc 14521
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 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-mre 14524  df-mrc 14525
This theorem is referenced by:  mrcidb2  14556
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