MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcidm Unicode version

Theorem mrcidm 13765
Description: The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcidm  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  ( F `  U ) )  =  ( F `  U
) )

Proof of Theorem mrcidm
StepHypRef Expression
1 mrcfval.f . . 3  |-  F  =  (mrCls `  C )
21mrccl 13757 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
31mrcid 13759 . 2  |-  ( ( C  e.  (Moore `  X )  /\  ( F `  U )  e.  C )  ->  ( F `  ( F `  U ) )  =  ( F `  U
) )
42, 3syldan 457 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  ( F `  U ) )  =  ( F `  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3257   ` cfv 5388  Moorecmre 13728  mrClscmrc 13729
This theorem is referenced by:  mrcuni  13767  mrcidmd  13772  ismrc  26440
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-fv 5396  df-mre 13732  df-mrc 13733
  Copyright terms: Public domain W3C validator