Theorem ntrval2 22914

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 4095	. . . . . 6 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		clscld.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval2 22913	. . . . . 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 4228	. . . . . . . 8 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	biimpi 216	. . . . . . 7 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
8	7	fveq2d 6844	. . . . . 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 4085	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
11	5, 10	eqtrd 2764	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
12	11	difeq2d 4085	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
13	2	ntropn 22912	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
14	2	eltopss 22770	. . . 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 4228	. . 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 2765	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ⊆ wss 3911  ∪ cuni 4867  ‘cfv 6499  Topctop 22756  intcnt 22880  clsccl 22881

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-top 22757  df-cld 22882  df-ntr 22883  df-cls 22884

This theorem is referenced by:  ntrdif  22915  ntrss  22918  kur14lem2  35167  dssmapntrcls  44090