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Theorem mrcid 16884
Description: The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcid ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)

Proof of Theorem mrcid
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 mress 16864 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → 𝑈𝑋)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcval 16881 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
41, 3syldan 594 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
5 intmin 4882 . . 3 (𝑈𝐶 {𝑠𝐶𝑈𝑠} = 𝑈)
65adantl 485 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → {𝑠𝐶𝑈𝑠} = 𝑈)
74, 6eqtrd 2859 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137  wss 3919   cint 4862  cfv 6343  Moorecmre 16853  mrClscmrc 16854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-mre 16857  df-mrc 16858
This theorem is referenced by:  mrcidb  16886  mrcidm  16890  mrcsscl  16891  isacs4lem  17778  dprdsn  19158  isnacs3  39567
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