Theorem clsint2 36330

Description: The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)

Hypothesis

Ref	Expression
clsint2.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsint2	⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐))

Distinct variable groups:   𝐶,𝑐   𝐽,𝑐   𝑋,𝑐

Proof of Theorem clsint2

Step	Hyp	Ref	Expression
1		sspwuni 5100	. . . 4 ⊢ (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 ⊆ 𝑋)
2		elssuni 4937	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐶 → 𝑐 ⊆ ∪ 𝐶)
3		sstr2 3990	. . . . . . . 8 ⊢ (𝑐 ⊆ ∪ 𝐶 → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
5	4	adantl 481	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
6		intss1 4963	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐶 → ∩ 𝐶 ⊆ 𝑐)
7		clsint2.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
8	7	clsss 23062	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ∩ 𝐶 ⊆ 𝑐) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
9	6, 8	syl3an3 1166	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ 𝑐 ∈ 𝐶) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
10	9	3com23 1127	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶 ∧ 𝑐 ⊆ 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11	10	3expia 1122	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (𝑐 ⊆ 𝑋 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
12	5, 11	syld 47	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (∪ 𝐶 ⊆ 𝑋 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
13	12	impancom 451	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐶 ⊆ 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
14	1, 13	sylan2b 594	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
15	14	ralrimiv 3145	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16		ssiin 5055	. 2 ⊢ (((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐) ↔ ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
17	15, 16	sylibr 234	1 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ∩ cint 4946  ∩ ciin 4992  ‘cfv 6561  Topctop 22899  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

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-cls 23029

This theorem is referenced by: (None)