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Theorem mrcsscl 13837
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 13810 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  V  C_  X )
213adant2 976 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  V  C_  X )
3 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
43mrcss 13833 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )
52, 4syld3an3 1229 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  ( F `  V
) )
63mrcid 13830 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  ( F `  V )  =  V )
763adant2 976 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  V )  =  V )
85, 7sseqtrd 3376 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800
This theorem is referenced by:  submrc  13845  isacs2  13870  isacs3lem  14584  mrelatlub  14604  gsumwspan  14783  cntzspan  15452  dprdspan  15577  subgdmdprd  15584  subgdprd  15585  dprdsn  15586  dprd2dlem1  15591  dprd2da  15592  dmdprdsplit2lem  15595  ablfac1b  15620  pgpfac1lem1  15624  pgpfac1lem5  15629  mrccss  16913  evlseu  19929  ismrcd2  26744  mrefg3  26753  isnacs3  26755  symggen  27379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-mre 13803  df-mrc 13804
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