Theorem clsint2 35818

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 5105	. . . 4 ⊢ (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 ⊆ 𝑋)
2		elssuni 4942	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐶 → 𝑐 ⊆ ∪ 𝐶)
3		sstr2 3987	. . . . . . . 8 ⊢ (𝑐 ⊆ ∪ 𝐶 → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
5	4	adantl 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
6		intss1 4968	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐶 → ∩ 𝐶 ⊆ 𝑐)
7		clsint2.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
8	7	clsss 22976	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ∩ 𝐶 ⊆ 𝑐) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
9	6, 8	syl3an3 1162	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ 𝑐 ∈ 𝐶) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
10	9	3com23 1123	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶 ∧ 𝑐 ⊆ 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11	10	3expia 1118	. . . . . 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 450	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐶 ⊆ 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
14	1, 13	sylan2b 592	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
15	14	ralrimiv 3141	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16		ssiin 5060	. 2 ⊢ (((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐) ↔ ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
17	15, 16	sylibr 233	1 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3057   ⊆ wss 3947  𝒫 cpw 4604  ∪ cuni 4910  ∩ cint 4951  ∩ ciin 4999  ‘cfv 6551  Topctop 22813  clsccl 22940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-iin 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-top 22814  df-cld 22941  df-cls 22943

This theorem is referenced by: (None)