MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcidb Structured version   Visualization version   GIF version

Theorem mrcidb 17660
Description: A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcidb (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))

Proof of Theorem mrcidb
StepHypRef Expression
1 mrcfval.f . . 3 𝐹 = (mrCls‘𝐶)
21mrcid 17658 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)
3 simpr 484 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) = 𝑈)
41mrcssv 17659 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
54adantr 480 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ⊆ 𝑋)
63, 5eqsstrrd 4035 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝑋)
71mrccl 17656 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)
86, 7syldan 591 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → (𝐹𝑈) ∈ 𝐶)
93, 8eqeltrrd 2840 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) = 𝑈) → 𝑈𝐶)
102, 9impbida 801 1 (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963  cfv 6563  Moorecmre 17627  mrClscmrc 17628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mre 17631  df-mrc 17632
This theorem is referenced by:  mrcidb2  17663
  Copyright terms: Public domain W3C validator