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Theorem mrcid 13531
Description: The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcid  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )

Proof of Theorem mrcid
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 mress 13511 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  U  C_  X )
2 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
32mrcval 13528 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
41, 3syldan 456 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
5 intmin 3898 . . 3  |-  ( U  e.  C  ->  |^| { s  e.  C  |  U  C_  s }  =  U )
65adantl 452 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  |^| { s  e.  C  |  U  C_  s }  =  U )
74, 6eqtrd 2328 1  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   |^|cint 3878   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  mrcidb  13533  mrcidm  13537  mrcsscl  13538  isacs4lem  14287  dprdsn  15287  isnacs3  26888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
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