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Theorem mrcsscl 16886
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 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → (𝐹𝑈) ⊆ 𝑉)

Proof of Theorem mrcsscl
StepHypRef Expression
1 mress 16859 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝐶) → 𝑉𝑋)
213adant2 1125 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → 𝑉𝑋)
3 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
43mrcss 16882 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))
52, 4syld3an3 1403 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → (𝐹𝑈) ⊆ (𝐹𝑉))
63mrcid 16879 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝐶) → (𝐹𝑉) = 𝑉)
763adant2 1125 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → (𝐹𝑉) = 𝑉)
85, 7sseqtrd 4011 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → (𝐹𝑈) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wss 3940  cfv 6354  Moorecmre 16848  mrClscmrc 16849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-mre 16852  df-mrc 16853
This theorem is referenced by:  submrc  16894  isacs2  16919  isacs3lem  17771  mrelatlub  17791  mndind  17987  gsumwspan  18006  symggen  18534  cntzspan  18900  dprdspan  19085  subgdmdprd  19092  subgdprd  19093  dprdsn  19094  dprd2dlem1  19099  dprd2da  19100  dmdprdsplit2lem  19103  ablfac1b  19128  pgpfac1lem1  19132  pgpfac1lem5  19137  evlseu  20231  mrccss  20773  ismrcd2  39180  mrefg3  39189  isnacs3  39191
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