Theorem clsint2 36312

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 4002	. . . . . . . 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 4968	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐶 → ∩ 𝐶 ⊆ 𝑐)
7		clsint2.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
8	7	clsss 23078	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ∩ 𝐶 ⊆ 𝑐) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
9	6, 8	syl3an3 1164	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ 𝑐 ∈ 𝐶) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
10	9	3com23 1125	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶 ∧ 𝑐 ⊆ 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11	10	3expia 1120	. . . . . 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 3143	. 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 234	1 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  ∩ cint 4951  ∩ ciin 4997  ‘cfv 6563  Topctop 22915  clsccl 23042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045

This theorem is referenced by: (None)