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Theorem kur14lem5 34189
Description: Lemma for kur14 34195. 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 22562 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴))
51, 2, 4mp2an 690 . 2 ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)
6 kur14lem.k . . 3 𝐾 = (cls‘𝐽)
76fveq1i 6889 . . 3 (𝐾𝐴) = ((cls‘𝐽)‘𝐴)
86, 7fveq12i 6894 . 2 (𝐾‘(𝐾𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴))
95, 8, 73eqtr4i 2770 1 (𝐾‘(𝐾𝐴)) = (𝐾𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wss 3947   cuni 4907  cfv 6540  Topctop 22386  intcnt 22512  clsccl 22513
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22387  df-cld 22514  df-cls 22516
This theorem is referenced by:  kur14lem6  34190  kur14lem7  34191
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