Theorem ntrval2 22993

Description: Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntrval2	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))

Proof of Theorem ntrval2

Step	Hyp	Ref	Expression
1		difss 4086	. . . . . 6 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		clscld.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval2 22992	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))))
4	1, 3	mpan2 691	. . . . 5 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))))
5	4	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))))
6		dfss4 4219	. . . . . . . 8 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	biimpi 216	. . . . . . 7 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
8	7	fveq2d 6836	. . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))) = ((int‘𝐽)‘𝑆))
9	8	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))) = ((int‘𝐽)‘𝑆))
10	9	difeq2d 4076	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
11	5, 10	eqtrd 2769	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
12	11	difeq2d 4076	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
13	2	ntropn 22991	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
14	2	eltopss 22849	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
15	13, 14	syldan 591	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16		dfss4 4219	. . 3 ⊢ (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
17	15, 16	sylib 218	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
18	12, 17	eqtr2d 2770	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3896   ⊆ wss 3899  ∪ cuni 4861  ‘cfv 6490  Topctop 22835  intcnt 22959  clsccl 22960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-top 22836  df-cld 22961  df-ntr 22962  df-cls 22963

This theorem is referenced by:  ntrdif  22994  ntrss  22997  kur14lem2  35350  dssmapntrcls  44311