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

Theorem mrcid 16481
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 16461 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → 𝑈𝑋)
2 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
32mrcval 16478 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
41, 3syldan 579 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
5 intmin 4631 . . 3 (𝑈𝐶 {𝑠𝐶𝑈𝑠} = 𝑈)
65adantl 467 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → {𝑠𝐶𝑈𝑠} = 𝑈)
74, 6eqtrd 2805 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065  wss 3723   cint 4611  cfv 6031  Moorecmre 16450  mrClscmrc 16451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mre 16454  df-mrc 16455
This theorem is referenced by:  mrcidb  16483  mrcidm  16487  mrcsscl  16488  isacs4lem  17376  dprdsn  18643  isnacs3  37799
  Copyright terms: Public domain W3C validator