Theorem clsun 36510

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 4230	. . . . . 6 ⊢ (𝑋 ∖ (𝐴 ∪ 𝐵)) = ((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))
2	1	fveq2i 6843	. . . . 5 ⊢ ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵)))
3		difss 4076	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
4		difss 4076	. . . . . . 7 ⊢ (𝑋 ∖ 𝐵) ⊆ 𝑋
5		clsun.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	ntrin 23026	. . . . . . 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 1456	. . . . . 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 2783	. . . 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 4130	. . . . . . 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 23017	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))))
15	10, 13, 14	syl2anc 585	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))))
16	5	ntrdif 23017	. . . . . . 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 23017	. . . . . . 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 4161	. . . . 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 4230	. . . . 5 ⊢ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
22	20, 21	eqtr4di 2789	. . . 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 2779	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
24	23	difeq2d 4066	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
25	5	clscld 23012	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽))
26	10, 13, 25	syl2anc 585	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽))
27	5	cldss 22994	. . . 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 4209	. . 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 23024	. . . . 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 23024	. . . . 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 4130	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37		dfss4 4209	. . . 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 2779	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 1542   ∈ wcel 2114   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  Clsdccld 22981  intcnt 22982  clsccl 22983

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-top 22859  df-cld 22984  df-ntr 22985  df-cls 22986

This theorem is referenced by: (None)