Theorem ntrdif 21354

Description: An interior of a complement is the complement of the closure. This set is also known as the exterior of 𝐴. (Contributed by Jeff Hankins, 31-Aug-2009.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

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

Proof of Theorem ntrdif

Step	Hyp	Ref	Expression
1		difss 3994	. . . 4 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
2		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	ntrval2 21353	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
4	1, 3	mpan2 678	. . 3 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
5	4	adantr 473	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
6		simpr 477	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
7		dfss4 4117	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
8	6, 7	sylib 210	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
9	8	fveq2d 6497	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))) = ((cls‘𝐽)‘𝐴))
10	9	difeq2d 3985	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
11	5, 10	eqtrd 2808	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048   ∖ cdif 3822   ⊆ wss 3825  ∪ cuni 4706  ‘cfv 6182  Topctop 21195  intcnt 21319  clsccl 21320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-top 21196  df-cld 21321  df-ntr 21322  df-cls 21323

This theorem is referenced by:  clsun  33137