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Theorem mrcsscl 14557
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 14530 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  V  C_  X )
213adant2 1007 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  V  C_  X )
3 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
43mrcss 14553 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )
52, 4syld3an3 1263 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  ( F `  V
) )
63mrcid 14550 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  e.  C )  ->  ( F `  V )  =  V )
763adant2 1007 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  V )  =  V )
85, 7sseqtrd 3391 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 965    = wceq 1369    e. wcel 1756    C_ wss 3327   ` cfv 5417  Moorecmre 14519  mrClscmrc 14520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-int 4128  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-mre 14523  df-mrc 14524
This theorem is referenced by:  submrc  14565  isacs2  14590  isacs3lem  15335  mrelatlub  15355  mrcmndind  15493  gsumwspan  15523  symggen  15975  cntzspan  16325  dprdspan  16523  subgdmdprd  16530  subgdprd  16531  dprdsn  16532  dprd2dlem1  16539  dprd2da  16540  dmdprdsplit2lem  16543  ablfac1b  16570  pgpfac1lem1  16574  pgpfac1lem5  16579  evlseu  17601  mrccss  18118  ismrcd2  29033  mrefg3  29042  isnacs3  29044
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