Theorem ntrval2 23029

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 4077	. . . . . 6 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		clscld.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval2 23028	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))))
4	1, 3	mpan2 692	. . . . 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 4210	. . . . . . . 8 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	biimpi 216	. . . . . . 7 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
8	7	fveq2d 6839	. . . . . 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 4067	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
11	5, 10	eqtrd 2772	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
12	11	difeq2d 4067	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
13	2	ntropn 23027	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
14	2	eltopss 22885	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
15	13, 14	syldan 592	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16		dfss4 4210	. . 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 2773	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ⊆ wss 3890  ∪ cuni 4851  ‘cfv 6493  Topctop 22871  intcnt 22995  clsccl 22996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22872  df-cld 22997  df-ntr 22998  df-cls 22999

This theorem is referenced by:  ntrdif  23030  ntrss  23033  kur14lem2  35408  dssmapntrcls  44576