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Theorem kur14lem2 31496
 Description: Lemma for kur14 31505. 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 21057 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴))))
51, 2, 4mp2an 710 . 2 ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴)))
6 kur14lem.i . . 3 𝐼 = (int‘𝐽)
76fveq1i 6353 . 2 (𝐼𝐴) = ((int‘𝐽)‘𝐴)
8 kur14lem.k . . . 4 𝐾 = (cls‘𝐽)
98fveq1i 6353 . . 3 (𝐾‘(𝑋𝐴)) = ((cls‘𝐽)‘(𝑋𝐴))
109difeq2i 3868 . 2 (𝑋 ∖ (𝐾‘(𝑋𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴)))
115, 7, 103eqtr4i 2792 1 (𝐼𝐴) = (𝑋 ∖ (𝐾‘(𝑋𝐴)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∈ wcel 2139   ∖ cdif 3712   ⊆ wss 3715  ∪ cuni 4588  ‘cfv 6049  Topctop 20900  intcnt 21023  clsccl 21024 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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-ntr 21026  df-cls 21027 This theorem is referenced by:  kur14lem6  31500  kur14lem7  31501
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