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Theorem kur14lem5 30446
 Description: Lemma for kur14 30452. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j 𝐽 ∈ Top
kur14lem.x 𝑋 = 𝐽
kur14lem.k 𝐾 = (cls‘𝐽)
kur14lem.i 𝐼 = (int‘𝐽)
kur14lem.a 𝐴𝑋
Assertion
Ref Expression
kur14lem5 (𝐾‘(𝐾𝐴)) = (𝐾𝐴)

Proof of Theorem kur14lem5
StepHypRef Expression
1 kur14lem.j . . 3 𝐽 ∈ Top
2 kur14lem.a . . 3 𝐴𝑋
3 kur14lem.x . . . 4 𝑋 = 𝐽
43clsidm 20681 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴))
51, 2, 4mp2an 704 . 2 ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)
6 kur14lem.k . . 3 𝐾 = (cls‘𝐽)
76fveq1i 6104 . . 3 (𝐾𝐴) = ((cls‘𝐽)‘𝐴)
86, 7fveq12i 6108 . 2 (𝐾‘(𝐾𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴))
95, 8, 73eqtr4i 2642 1 (𝐾‘(𝐾𝐴)) = (𝐾𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  intcnt 20631  clsccl 20632 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-cls 20635 This theorem is referenced by:  kur14lem6  30447  kur14lem7  30448
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