Theorem clsint2 36295

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 5123	. . . 4 ⊢ (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 ⊆ 𝑋)
2		elssuni 4961	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐶 → 𝑐 ⊆ ∪ 𝐶)
3		sstr2 4015	. . . . . . . 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 4987	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐶 → ∩ 𝐶 ⊆ 𝑐)
7		clsint2.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
8	7	clsss 23083	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ∩ 𝐶 ⊆ 𝑐) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
9	6, 8	syl3an3 1165	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ 𝑐 ∈ 𝐶) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
10	9	3com23 1126	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶 ∧ 𝑐 ⊆ 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11	10	3expia 1121	. . . . . 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 593	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
15	14	ralrimiv 3151	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16		ssiin 5078	. 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 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  ∩ cint 4970  ∩ ciin 5016  ‘cfv 6573  Topctop 22920  clsccl 23047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-cld 23048  df-cls 23050

This theorem is referenced by: (None)