Theorem clsint2 36702

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 5062	. . . 4 ⊢ (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 ⊆ 𝑋)
2		elssuni 4900	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐶 → 𝑐 ⊆ ∪ 𝐶)
3		sstr2 3946	. . . . . . . 8 ⊢ (𝑐 ⊆ ∪ 𝐶 → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
4	2, 3	syl 18	. . . . . . 7 ⊢ (𝑐 ∈ 𝐶 → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
5	4	adantl 486	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (∪ 𝐶 ⊆ 𝑋 → 𝑐 ⊆ 𝑋))
6		intss1 4924	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐶 → ∩ 𝐶 ⊆ 𝑐)
7		clsint2.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
8	7	clsss 23172	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ∩ 𝐶 ⊆ 𝑐) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
9	6, 8	syl3an3 1181	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ 𝑐 ∈ 𝐶) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
10	9	3com23 1142	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶 ∧ 𝑐 ⊆ 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
11	10	3expia 1137	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (𝑐 ⊆ 𝑋 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
12	5, 11	syld 48	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ∈ 𝐶) → (∪ 𝐶 ⊆ 𝑋 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
13	12	impancom 456	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐶 ⊆ 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
14	1, 13	sylan2b 605	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → (𝑐 ∈ 𝐶 → ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐)))
15	14	ralrimiv 3156	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
16		ssiin 5016	. 2 ⊢ (((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐) ↔ ∀𝑐 ∈ 𝐶 ((cls‘𝐽)‘∩ 𝐶) ⊆ ((cls‘𝐽)‘𝑐))
17	15, 16	sylibr 237	1 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘∩ 𝐶) ⊆ ∩ 𝑐 ∈ 𝐶 ((cls‘𝐽)‘𝑐))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ⊆ wss 3907  𝒫 cpw 4558  ∪ cuni 4868  ∩ cint 4908  ∩ ciin 4953  ‘cfv 6525  Topctop 23011  clsccl 23136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-top 23012  df-cld 23137  df-cls 23139

This theorem is referenced by: (None)