Theorem clsval 23316

Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsval	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem clsval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	clsfval 23304	. . . 4 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}))
3	2	fveq1d 6875	. . 3 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆))
4	3	adantr 486	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆))
5		eqid 2760	. . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})
6		sseq1 3955	. . . . 5 ⊢ (𝑦 = 𝑆 → (𝑦 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑥))
7	6	rabbidv 3419	. . . 4 ⊢ (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
8	7	inteqd 4911	. . 3 ⊢ (𝑦 = 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
9	1	topopn 23185	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
10		elpw2g 5294	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
11	9, 10	syl 18	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
12	11	biimpar 483	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
13	1	topcld 23314	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
14		sseq2 3956	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
15	14	rspcev 3576	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥)
16	13, 15	sylan 592	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥)
17		intexrab 5307	. . . 4 ⊢ (∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V)
18	16, 17	sylib 221	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V)
19	5, 8, 12, 18	fvmptd3 7005	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
20	4, 19	eqtrd 2795	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3898  𝒫 cpw 4556  ∪ cuni 4866  ∩ cint 4906   ↦ cmpt 5185  ‘cfv 6527  Topctop 23172  Clsdccld 23295  clsccl 23297

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-top 23173  df-cld 23298  df-cls 23300

This theorem is used by:  cldcls  23321  clscld  23326  clsf  23327  clsval2  23329  clsss  23333  sscls  23335