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Theorem mrcidb2 13802
Description: A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcidb2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U )  C_  U
) )

Proof of Theorem mrcidb2
StepHypRef Expression
1 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
21mrcidb 13799 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
32adantr 452 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
41mrcssid 13801 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U
) )
54biantrud 494 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  (
( F `  U
)  C_  U  <->  ( ( F `  U )  C_  U  /\  U  C_  ( F `  U ) ) ) )
6 eqss 3327 . . 3  |-  ( ( F `  U )  =  U  <->  ( ( F `  U )  C_  U  /\  U  C_  ( F `  U ) ) )
75, 6syl6rbbr 256 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  (
( F `  U
)  =  U  <->  ( F `  U )  C_  U
) )
83, 7bitrd 245 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U )  C_  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3284   ` cfv 5417  Moorecmre 13766  mrClscmrc 13767
This theorem is referenced by:  isacs5  14557
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-mre 13770  df-mrc 13771
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