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Theorem mrcsscl 13522
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 13495 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  V  C_  X )
213adant2 974 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  V  C_  X )
3 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
43mrcss 13518 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )
52, 4syld3an3 1227 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  ( F `  V
) )
63mrcid 13515 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  ( F `  V )  =  V )
763adant2 974 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  V )  =  V )
85, 7sseqtrd 3214 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 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  submrc  13530  isacs2  13555  isacs3lem  14269  mrelatlub  14289  gsumwspan  14468  cntzspan  15137  dprdspan  15262  subgdmdprd  15269  subgdprd  15270  dprdsn  15271  dprd2dlem1  15276  dprd2da  15277  dmdprdsplit2lem  15280  ablfac1b  15305  pgpfac1lem1  15309  pgpfac1lem5  15314  mrccss  16594  evlseu  19400  ismrcd2  26774  mrefg3  26783  isnacs3  26785  symggen  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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