Theorem kur14lem2 35738

Description: Lemma for kur14 35747. 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 23260	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))))
5	1, 2, 4	mp2an 705	. 2 ⊢ ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))
6		kur14lem.i	. . 3 ⊢ 𝐼 = (int‘𝐽)
7	6	fveq1i 6886	. 2 ⊢ (𝐼‘𝐴) = ((int‘𝐽)‘𝐴)
8		kur14lem.k	. . . 4 ⊢ 𝐾 = (cls‘𝐽)
9	8	fveq1i 6886	. . 3 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = ((cls‘𝐽)‘(𝑋 ∖ 𝐴))
10	9	difeq2i 4078	. 2 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))
11	5, 7, 10	3eqtr4i 2798	1 ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4874  ‘cfv 6540  Topctop 23102  intcnt 23226  clsccl 23227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 23103  df-cld 23228  df-ntr 23229  df-cls 23230

This theorem is used by:  kur14lem6  35742  kur14lem7  35743