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

Theorem mrcsscl 14892
Description: The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcsscl  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  V )

Proof of Theorem mrcsscl
StepHypRef Expression
1 mress 14865 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  V  C_  X )
213adant2 1015 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  V  C_  X )
3 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
43mrcss 14888 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )
52, 4syld3an3 1273 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  ( F `  V
) )
63mrcid 14885 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  ( F `  V )  =  V )
763adant2 1015 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  V )  =  V )
85, 7sseqtrd 3545 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   ` cfv 5594  Moorecmre 14854  mrClscmrc 14855
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 14858  df-mrc 14859
This theorem is referenced by:  submrc  14900  isacs2  14925  isacs3lem  15670  mrelatlub  15690  mrcmndind  15869  gsumwspan  15886  symggen  16368  cntzspan  16723  dprdspan  16946  subgdmdprd  16953  subgdprd  16954  dprdsn  16955  dprd2dlem1  16962  dprd2da  16963  dmdprdsplit2lem  16966  ablfac1b  16993  pgpfac1lem1  16997  pgpfac1lem5  17002  evlseu  18055  mrccss  18594  ismrcd2  30559  mrefg3  30568  isnacs3  30570
  Copyright terms: Public domain W3C validator