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Theorem mrcsscl 13571
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 13544 . . . 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 13567 . . 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 13564 . . 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 3248 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 1633    e. wcel 1701    C_ wss 3186   ` cfv 5292  Moorecmre 13533  mrClscmrc 13534
This theorem is referenced by:  submrc  13579  isacs2  13604  isacs3lem  14318  mrelatlub  14338  gsumwspan  14517  cntzspan  15186  dprdspan  15311  subgdmdprd  15318  subgdprd  15319  dprdsn  15320  dprd2dlem1  15325  dprd2da  15326  dmdprdsplit2lem  15329  ablfac1b  15354  pgpfac1lem1  15358  pgpfac1lem5  15363  mrccss  16650  evlseu  19453  ismrcd2  25922  mrefg3  25931  isnacs3  25933  symggen  26559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-mre 13537  df-mrc 13538
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