Theorem clsval 20589

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 20577	. . . 4 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}))
3	2	fveq1d 6086	. . 3 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆))
4	3	adantr 479	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆))
5	1	topopn 20474	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6		elpw2g 4745	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
7	5, 6	syl 17	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
8	7	biimpar 500	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
9	1	topcld 20587	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
10		sseq2 3585	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
11	10	rspcev 3277	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥)
12	9, 11	sylan 486	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥)
13		intexrab 4741	. . . 4 ⊢ (∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V)
14	12, 13	sylib 206	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V)
15		sseq1 3584	. . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑥))
16	15	rabbidv 3159	. . . . 5 ⊢ (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
17	16	inteqd 4405	. . . 4 ⊢ (𝑦 = 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
18		eqid 2605	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})
19	17, 18	fvmptg 6170	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
20	8, 14, 19	syl2anc 690	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
21	4, 20	eqtrd 2639	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})

Syntax hints:   → wi 4   ↔ wb 194   ∧ wa 382   = wceq 1474   ∈ wcel 1975  ∃wrex 2892  {crab 2895  Vcvv 3168   ⊆ wss 3535  𝒫 cpw 4103  ∪ cuni 4362  ∩ cint 4400   ↦ cmpt 4633  ‘cfv 5786  Topctop 20455  Clsdccld 20568  clsccl 20570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-top 20459  df-cld 20571  df-cls 20573

This theorem is referenced by:  cldcls  20594  clscld  20599  clsf  20600  clsval2  20602  clsss  20606  sscls  20608