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Theorem kur14lem5 31521
Description: Lemma for kur14 31527. 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 21094 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴))
51, 2, 4mp2an 710 . 2 ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)
6 kur14lem.k . . 3 𝐾 = (cls‘𝐽)
76fveq1i 6355 . . 3 (𝐾𝐴) = ((cls‘𝐽)‘𝐴)
86, 7fveq12i 6359 . 2 (𝐾‘(𝐾𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴))
95, 8, 73eqtr4i 2793 1 (𝐾‘(𝐾𝐴)) = (𝐾𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2140  wss 3716   cuni 4589  cfv 6050  Topctop 20921  intcnt 21044  clsccl 21045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-top 20922  df-cld 21046  df-cls 21048
This theorem is referenced by:  kur14lem6  31522  kur14lem7  31523
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