Theorem clsun 36316

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 4253	. . . . . 6 ⊢ (𝑋 ∖ (𝐴 ∪ 𝐵)) = ((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))
2	1	fveq2i 6861	. . . . 5 ⊢ ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵)))
3		difss 4099	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
4		difss 4099	. . . . . . 7 ⊢ (𝑋 ∖ 𝐵) ⊆ 𝑋
5		clsun.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	ntrin 22948	. . . . . . 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 1133	. . . . 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 2776	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
10		simp1 1136	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐽 ∈ Top)
11		unss 4153	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
12	11	biimpi 216	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
13	12	3adant1 1130	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
14	5	ntrdif 22939	. . . . 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 22939	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17	16	3adant3 1132	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
18	5	ntrdif 22939	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19	18	3adant2 1131	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
20	17, 19	ineq12d 4184	. . . . 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 4253	. . . . 5 ⊢ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
22	20, 21	eqtr4di 2782	. . . 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 2772	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
24	23	difeq2d 4089	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
25	5	clscld 22934	. . . . 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 22916	. . . 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 4232	. . 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 22946	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32	31	3adant3 1132	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
33	5	clsss3 22946	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34	33	3adant2 1131	. . . 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 4153	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37		dfss4 4232	. . . 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 2772	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  Clsdccld 22903  intcnt 22904  clsccl 22905

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-cld 22906  df-ntr 22907  df-cls 22908

This theorem is referenced by: (None)