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Theorem kur14lem2 32528
Description: Lemma for kur14 32537. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (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
kur14lem2 (𝐼𝐴) = (𝑋 ∖ (𝐾‘(𝑋𝐴)))

Proof of Theorem kur14lem2
StepHypRef Expression
1 kur14lem.j . . 3 𝐽 ∈ Top
2 kur14lem.a . . 3 𝐴𝑋
3 kur14lem.x . . . 4 𝑋 = 𝐽
43ntrval2 21654 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴))))
51, 2, 4mp2an 691 . 2 ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴)))
6 kur14lem.i . . 3 𝐼 = (int‘𝐽)
76fveq1i 6653 . 2 (𝐼𝐴) = ((int‘𝐽)‘𝐴)
8 kur14lem.k . . . 4 𝐾 = (cls‘𝐽)
98fveq1i 6653 . . 3 (𝐾‘(𝑋𝐴)) = ((cls‘𝐽)‘(𝑋𝐴))
109difeq2i 4071 . 2 (𝑋 ∖ (𝐾‘(𝑋𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴)))
115, 7, 103eqtr4i 2855 1 (𝐼𝐴) = (𝑋 ∖ (𝐾‘(𝑋𝐴)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  cdif 3905  wss 3908   cuni 4813  cfv 6334  Topctop 21496  intcnt 21620  clsccl 21621
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-top 21497  df-cld 21622  df-ntr 21623  df-cls 21624
This theorem is referenced by:  kur14lem6  32532  kur14lem7  32533
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