Theorem kur14lem2 35192

Description: Lemma for kur14 35201. 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	⊢ 𝐴 ⊆ 𝑋

Assertion

Ref	Expression
kur14lem2	⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))

Proof of Theorem kur14lem2

Step	Hyp	Ref	Expression
1		kur14lem.j	. . 3 ⊢ 𝐽 ∈ Top
2		kur14lem.a	. . 3 ⊢ 𝐴 ⊆ 𝑋
3		kur14lem.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
4	3	ntrval2 23075	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))))
5	1, 2, 4	mp2an 692	. 2 ⊢ ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))
6		kur14lem.i	. . 3 ⊢ 𝐼 = (int‘𝐽)
7	6	fveq1i 6908	. 2 ⊢ (𝐼‘𝐴) = ((int‘𝐽)‘𝐴)
8		kur14lem.k	. . . 4 ⊢ 𝐾 = (cls‘𝐽)
9	8	fveq1i 6908	. . 3 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = ((cls‘𝐽)‘(𝑋 ∖ 𝐴))
10	9	difeq2i 4133	. 2 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))
11	5, 7, 10	3eqtr4i 2773	1 ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))

Syntax hints:   = wceq 1537   ∈ wcel 2106   ∖ cdif 3960   ⊆ wss 3963  ∪ cuni 4912  ‘cfv 6563  Topctop 22915  intcnt 23041  clsccl 23042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-ntr 23044  df-cls 23045

This theorem is referenced by:  kur14lem6  35196  kur14lem7  35197