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Theorem kur14lem5 34500
Description: Lemma for kur14 34506. 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 = βˆͺ 𝐽
kur14lem.k 𝐾 = (clsβ€˜π½)
kur14lem.i 𝐼 = (intβ€˜π½)
kur14lem.a 𝐴 βŠ† 𝑋
Assertion
Ref Expression
kur14lem5 (πΎβ€˜(πΎβ€˜π΄)) = (πΎβ€˜π΄)

Proof of Theorem kur14lem5
StepHypRef Expression
1 kur14lem.j . . 3 𝐽 ∈ Top
2 kur14lem.a . . 3 𝐴 βŠ† 𝑋
3 kur14lem.x . . . 4 𝑋 = βˆͺ 𝐽
43clsidm 22792 . . 3 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = ((clsβ€˜π½)β€˜π΄))
51, 2, 4mp2an 689 . 2 ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = ((clsβ€˜π½)β€˜π΄)
6 kur14lem.k . . 3 𝐾 = (clsβ€˜π½)
76fveq1i 6892 . . 3 (πΎβ€˜π΄) = ((clsβ€˜π½)β€˜π΄)
86, 7fveq12i 6897 . 2 (πΎβ€˜(πΎβ€˜π΄)) = ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄))
95, 8, 73eqtr4i 2769 1 (πΎβ€˜(πΎβ€˜π΄)) = (πΎβ€˜π΄)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22616  intcnt 22742  clsccl 22743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22617  df-cld 22744  df-cls 22746
This theorem is referenced by:  kur14lem6  34501  kur14lem7  34502
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