Theorem clsval 14963

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 14953	. . . 4 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}))
3	2	fveq1d 5671	. . 3 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆))
4	3	adantr 276	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆))
5		eqid 2232	. . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})
6		sseq1 3260	. . . . 5 ⊢ (𝑦 = 𝑆 → (𝑦 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑥))
7	6	rabbidv 2801	. . . 4 ⊢ (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
8	7	inteqd 3953	. . 3 ⊢ (𝑦 = 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
9	1	topopn 14860	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
10		elpw2g 4267	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
11	9, 10	syl 14	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
12	11	biimpar 297	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
13	1	topcld 14961	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
14		sseq2 3261	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
15	14	rspcev 2920	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥)
16	13, 15	sylan 283	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥)
17		intexrabim 4264	. . . 4 ⊢ (∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V)
18	16, 17	syl 14	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V)
19	5, 8, 12, 18	fvmptd3 5770	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
20	4, 19	eqtrd 2265	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  {crab 2524  Vcvv 2812   ⊆ wss 3210  𝒫 cpw 3668  ∪ cuni 3913  ∩ cint 3948   ↦ cmpt 4170  ‘cfv 5351  Topctop 14849  Clsdccld 14944  clsccl 14946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-top 14850  df-cld 14947  df-cls 14949

This theorem is referenced by:  cldcls  14966  clsss  14970  sscls  14972