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Theorem mrcssv 13831
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 5746 . 2  |-  ( F `
 U )  C_  U.
ran  F
2 mrcfval.f . . . . 5  |-  F  =  (mrCls `  C )
32mrcf 13826 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
4 frn 5589 . . . 4  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
5 uniss 4028 . . . 4  |-  ( ran 
F  C_  C  ->  U.
ran  F  C_  U. C
)
63, 4, 53syl 19 . . 3  |-  ( C  e.  (Moore `  X
)  ->  U. ran  F  C_ 
U. C )
7 mreuni 13817 . . 3  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
86, 7sseqtrd 3376 . 2  |-  ( C  e.  (Moore `  X
)  ->  U. ran  F  C_  X )
91, 8syl5ss 3351 1  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U )  C_  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   ran crn 4871   -->wf 5442   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800
This theorem is referenced by:  mrcidb  13832  mrcuni  13838  mrcssvd  13840  mrefg2  26752  proot1hash  27487
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-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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