Theorem cldbnd 36539

Description: A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)

Hypothesis

Ref	Expression
opnbnd.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldbnd	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))

Proof of Theorem cldbnd

Step	Hyp	Ref	Expression
1		opnbnd.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	iscld3 23020	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3		eqimss 3994	. . . 4 ⊢ (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
4	2, 3	biimtrdi 253	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5		ssinss1 4200	. . 3 ⊢ (((cls‘𝐽)‘𝐴) ⊆ 𝐴 → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴)
6	4, 5	syl6 35	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))
7		sslin 4197	. . . . . 6 ⊢ ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴))
8	7	adantl 481	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴))
9		disjdifr 4427	. . . . 5 ⊢ ((𝑋 ∖ 𝐴) ∩ 𝐴) = ∅
10		sseq0 4357	. . . . 5 ⊢ ((((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴) ∧ ((𝑋 ∖ 𝐴) ∩ 𝐴) = ∅) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅)
11	8, 9, 10	sylancl 587	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅)
12	11	ex 412	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅))
13		incom 4163	. . . . . . . 8 ⊢ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘𝐴))
14		dfss4 4223	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		fveq2 6842	. . . . . . . . . . . 12 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))) = ((cls‘𝐽)‘𝐴))
16	15	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
17	14, 16	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
19	18	ineq2d 4174	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
20	13, 19	eqtrid 2784	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
21	20	ineq2d 4174	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))))
22	21	eqeq1d 2739	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
23		difss 4090	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
24	1	opnbnd 36538	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
25	23, 24	mpan2 692	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
26	25	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
27	22, 26	bitr4d 282	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
28	1	opncld 22989	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽))
29	28	ex 412	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽)))
30	29	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽)))
31		eleq1 2825	. . . . . . 7 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
32	14, 31	sylbi 217	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
33	32	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
34	30, 33	sylibd 239	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 → 𝐴 ∈ (Clsd‘𝐽)))
35	27, 34	sylbid 240	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ → 𝐴 ∈ (Clsd‘𝐽)))
36	12, 35	syld 47	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → 𝐴 ∈ (Clsd‘𝐽)))
37	6, 36	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865  ‘cfv 6500  Topctop 22849  Clsdccld 22972  clsccl 22974

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977

This theorem is referenced by: (None)