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Theorem mrcid 14547
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 14527 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  U  C_  X )
2 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
32mrcval 14544 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
41, 3syldan 467 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
5 intmin 4145 . . 3  |-  ( U  e.  C  ->  |^| { s  e.  C  |  U  C_  s }  =  U )
65adantl 463 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  |^| { s  e.  C  |  U  C_  s }  =  U )
74, 6eqtrd 2473 1  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   {crab 2717    C_ wss 3325   |^|cint 4125   ` cfv 5415  Moorecmre 14516  mrClscmrc 14517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-mre 14520  df-mrc 14521
This theorem is referenced by:  mrcidb  14549  mrcidm  14553  mrcsscl  14554  isacs4lem  15334  dprdsn  16523  isnacs3  28971
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