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Theorem mrcidb2 14869
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 14866 . . 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 14868 . . . 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 3519 . . 3  |-  ( ( F `  U )  =  U  <->  ( ( F `  U )  C_  U  /\  U  C_  ( F `  U ) ) )
75, 6syl6rbbr 264 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  (
( F `  U
)  =  U  <->  ( F `  U )  C_  U
) )
83, 7bitrd 253 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 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   ` cfv 5586  Moorecmre 14833  mrClscmrc 14834
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-mre 14837  df-mrc 14838
This theorem is referenced by:  isacs5  15655
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