Theorem clsun 36329

Description: A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)

Hypothesis

Ref	Expression
clsun.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsun	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))

Proof of Theorem clsun

Step	Hyp	Ref	Expression
1		difundi 4290	. . . . . 6 ⊢ (𝑋 ∖ (𝐴 ∪ 𝐵)) = ((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))
2	1	fveq2i 6909	. . . . 5 ⊢ ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵)))
3		difss 4136	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
4		difss 4136	. . . . . . 7 ⊢ (𝑋 ∖ 𝐵) ⊆ 𝑋
5		clsun.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	ntrin 23069	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋 ∧ (𝑋 ∖ 𝐵) ⊆ 𝑋) → ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
7	3, 4, 6	mp3an23 1455	. . . . . 6 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
8	7	3ad2ant1 1134	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
9	2, 8	eqtrid 2789	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
10		simp1 1137	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐽 ∈ Top)
11		unss 4190	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
12	11	biimpi 216	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
13	12	3adant1 1131	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
14	5	ntrdif 23060	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))))
15	10, 13, 14	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))))
16	5	ntrdif 23060	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17	16	3adant3 1133	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
18	5	ntrdif 23060	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19	18	3adant2 1132	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
20	17, 19	ineq12d 4221	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵))))
21		difundi 4290	. . . . 5 ⊢ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
22	20, 21	eqtr4di 2795	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
23	9, 15, 22	3eqtr3d 2785	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
24	23	difeq2d 4126	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
25	5	clscld 23055	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽))
26	10, 13, 25	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽))
27	5	cldss 23037	. . . 4 ⊢ (((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ⊆ 𝑋)
28	26, 27	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ⊆ 𝑋)
29		dfss4 4269	. . 3 ⊢ (((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))
30	28, 29	sylib 218	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))
31	5	clsss3 23067	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32	31	3adant3 1133	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
33	5	clsss3 23067	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34	33	3adant2 1132	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
35	32, 34	jca 511	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36		unss 4190	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37		dfss4 4269	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
38	36, 37	bitri 275	. . 3 ⊢ ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
39	35, 38	sylib 218	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))
40	24, 30, 39	3eqtr3d 2785	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  Clsdccld 23024  intcnt 23025  clsccl 23026

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029

This theorem is referenced by: (None)