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Theorem mrcssid 16878
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 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ⊆ (𝐹𝑈))

Proof of Theorem mrcssid
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4892 . 2 𝑈 {𝑠𝐶𝑈𝑠}
2 mrcfval.f . . 3 𝐹 = (mrCls‘𝐶)
32mrcval 16871 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
41, 3sseqtrrid 4024 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ⊆ (𝐹𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {crab 3147  wss 3940   cint 4874  cfv 6352  Moorecmre 16843  mrClscmrc 16844
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 7451
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 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-mre 16847  df-mrc 16848
This theorem is referenced by:  mrcidb2  16879  mrcuni  16882  mrcssidd  16886  mrelatlub  17786  gsumwspan  17994  symggen  18518  mrccss  20754  ismrcd2  39161  ismrc  39163  mrefg2  39169
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