Theorem clsun 36955

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 4239	. . . . . 6 ⊢ (𝑋 ∖ (𝐴 ∪ 𝐵)) = ((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))
2	1	fveq2i 6885	. . . . 5 ⊢ ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵)))
3		difss 4086	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
4		difss 4086	. . . . . . 7 ⊢ (𝑋 ∖ 𝐵) ⊆ 𝑋
5		clsun.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	ntrin 23292	. . . . . . 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 1482	. . . . . 6 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘((𝑋 ∖ 𝐴) ∩ (𝑋 ∖ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
8	7	3ad2ant1 1151	. . . . 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 2809	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (((int‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((int‘𝐽)‘(𝑋 ∖ 𝐵))))
10		simp1 1154	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐽 ∈ Top)
11		unss 4139	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
12	11	biimpi 219	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
13	12	3adant1 1148	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
14	5	ntrdif 23283	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))))
15	10, 13, 14	syl2anc 596	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝐴 ∪ 𝐵))) = (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))))
16	5	ntrdif 23283	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
17	16	3adant3 1150	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
18	5	ntrdif 23283	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
19	18	3adant2 1149	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐵)) = (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
20	17, 19	ineq12d 4170	. . . . 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 4239	. . . . 5 ⊢ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))) = ((𝑋 ∖ ((cls‘𝐽)‘𝐴)) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝐵)))
22	20, 21	eqtr4di 2815	. . . 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 2805	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵))))
24	23	difeq2d 4077	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = (𝑋 ∖ (𝑋 ∖ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))))
25	5	clscld 23278	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽))
26	10, 13, 25	syl2anc 596	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽))
27	5	cldss 23260	. . . 4 ⊢ (((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ⊆ 𝑋)
28	26, 27	syl 18	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ⊆ 𝑋)
29		dfss4 4218	. . 3 ⊢ (((cls‘𝐽)‘(𝐴 ∪ 𝐵)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))
30	28, 29	sylib 221	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))) = ((cls‘𝐽)‘(𝐴 ∪ 𝐵)))
31	5	clsss3 23290	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
32	31	3adant3 1150	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
33	5	clsss3 23290	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
34	33	3adant2 1149	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘𝐵) ⊆ 𝑋)
35	32, 34	jca 521	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋))
36		unss 4139	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ⊆ 𝑋 ∧ ((cls‘𝐽)‘𝐵) ⊆ 𝑋) ↔ (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)) ⊆ 𝑋)
37		dfss4 4218	. . . 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 278	. . 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 221	. 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 2805	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((cls‘𝐽)‘(𝐴 ∪ 𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4870  ‘cfv 6537  Topctop 23124  Clsdccld 23247  intcnt 23248  clsccl 23249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23125  df-cld 23250  df-ntr 23251  df-cls 23252

This theorem is used by: (None)