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Theorem kur14lem2 34129
Description: Lemma for kur14 34138. 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 22524 . . 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 6882 . 2 (𝐼𝐴) = ((int‘𝐽)‘𝐴)
8 kur14lem.k . . . 4 𝐾 = (cls‘𝐽)
98fveq1i 6882 . . 3 (𝐾‘(𝑋𝐴)) = ((cls‘𝐽)‘(𝑋𝐴))
109difeq2i 4117 . 2 (𝑋 ∖ (𝐾‘(𝑋𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝐴)))
115, 7, 103eqtr4i 2771 1 (𝐼𝐴) = (𝑋 ∖ (𝐾‘(𝑋𝐴)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  cdif 3943  wss 3946   cuni 4904  cfv 6535  Topctop 22364  intcnt 22490  clsccl 22491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-top 22365  df-cld 22492  df-ntr 22493  df-cls 22494
This theorem is referenced by:  kur14lem6  34133  kur14lem7  34134
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