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Theorem mrcssv 13767
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 5695 . 2  |-  ( F `
 U )  C_  U.
ran  F
2 mrcfval.f . . . . 5  |-  F  =  (mrCls `  C )
32mrcf 13762 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
4 frn 5538 . . . 4  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
5 uniss 3979 . . . 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 13753 . . 3  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
86, 7sseqtrd 3328 . 2  |-  ( C  e.  (Moore `  X
)  ->  U. ran  F  C_  X )
91, 8syl5ss 3303 1  |-  ( C  e.  (Moore `  X
)  ->  ( F `  U )  C_  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3264   ~Pcpw 3743   U.cuni 3958   ran crn 4820   -->wf 5391   ` cfv 5395  Moorecmre 13735  mrClscmrc 13736
This theorem is referenced by:  mrcidb  13768  mrcuni  13774  mrcssvd  13776  mrefg2  26453  proot1hash  27189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-mre 13739  df-mrc 13740
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