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Theorem mrcssv 15513
Description: The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcssv  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U )  C_  X
)

Proof of Theorem mrcssv
StepHypRef Expression
1 fvssunirn 5902 . 2  |-  ( F `
 U )  C_  U.
ran  F
2 mrcfval.f . . . . 5  |-  F  =  (mrCls `  C )
32mrcf 15508 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
4 frn 5750 . . . 4  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
5 uniss 4238 . . . 4  |-  ( ran 
F  C_  C  ->  U.
ran  F  C_  U. C
)
63, 4, 53syl 18 . . 3  |-  ( C  e.  (Moore `  X
)  ->  U. ran  F  C_ 
U. C )
7 mreuni 15499 . . 3  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
86, 7sseqtrd 3501 . 2  |-  ( C  e.  (Moore `  X
)  ->  U. ran  F  C_  X )
91, 8syl5ss 3476 1  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U )  C_  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869    C_ wss 3437   ~Pcpw 3980   U.cuni 4217   ran crn 4852   -->wf 5595   ` cfv 5599  Moorecmre 15481  mrClscmrc 15482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-int 4254  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fv 5607  df-mre 15485  df-mrc 15486
This theorem is referenced by:  mrcidb  15514  mrcuni  15520  mrcssvd  15522  mrefg2  35512  proot1hash  36041
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