Theorem kur14lem2 35194

Description: Lemma for kur14 35203. 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 22938	. . 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 6859	. 2 ⊢ (𝐼‘𝐴) = ((int‘𝐽)‘𝐴)
8		kur14lem.k	. . . 4 ⊢ 𝐾 = (cls‘𝐽)
9	8	fveq1i 6859	. . 3 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = ((cls‘𝐽)‘(𝑋 ∖ 𝐴))
10	9	difeq2i 4086	. 2 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))
11	5, 7, 10	3eqtr4i 2762	1 ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))

Syntax hints:   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911   ⊆ wss 3914  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  intcnt 22904  clsccl 22905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-cld 22906  df-ntr 22907  df-cls 22908

This theorem is referenced by:  kur14lem6  35198  kur14lem7  35199