Theorem clsdif 22956

Description: A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

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

Proof of Theorem clsdif

Step	Hyp	Ref	Expression
1		difss 4089	. . . 4 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
2		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval2 22953	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
4	1, 3	mpan2 691	. . 3 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
5	4	adantr 480	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
6		simpr 484	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
7		dfss4 4222	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
8	6, 7	sylib 218	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
9	8	fveq2d 6830	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))) = ((int‘𝐽)‘𝐴))
10	9	difeq2d 4079	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
11	5, 10	eqtrd 2764	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902   ⊆ wss 3905  ∪ cuni 4861  ‘cfv 6486  Topctop 22796  intcnt 22920  clsccl 22921

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-top 22797  df-cld 22922  df-ntr 22923  df-cls 22924

This theorem is referenced by:  maxlp  23050  topbnd  36297