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Theorem mrcssid 14576
Description: The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcssid  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U
) )

Proof of Theorem mrcssid
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ssintub 4167 . 2  |-  U  C_  |^|
{ s  e.  C  |  U  C_  s }
2 mrcfval.f . . 3  |-  F  =  (mrCls `  C )
32mrcval 14569 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
41, 3syl5sseqr 3426 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2740    C_ wss 3349   |^|cint 4149   ` cfv 5439  Moorecmre 14541  mrClscmrc 14542
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 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-mre 14545  df-mrc 14546
This theorem is referenced by:  mrcidb2  14577  mrcuni  14580  mrcssidd  14584  mrelatlub  15377  gsumwspan  15545  symggen  15997  mrccss  18141  ismrcd2  29061  ismrc  29063  mrefg2  29069
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