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Theorem mrcidb2 15234
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 15231 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
32adantr 465 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
41mrcssid 15233 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U
) )
54biantrud 507 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  (
( F `  U
)  C_  U  <->  ( ( F `  U )  C_  U  /\  U  C_  ( F `  U ) ) ) )
6 eqss 3459 . . 3  |-  ( ( F `  U )  =  U  <->  ( ( F `  U )  C_  U  /\  U  C_  ( F `  U ) ) )
75, 6syl6rbbr 266 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  (
( F `  U
)  =  U  <->  ( F `  U )  C_  U
) )
83, 7bitrd 255 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U )  C_  U
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844    C_ wss 3416   ` cfv 5571  Moorecmre 15198  mrClscmrc 15199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-mre 15202  df-mrc 15203
This theorem is referenced by:  isacs5  16128
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