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Theorem clsval 22892
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 ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem clsval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5 𝑋 = 𝐽
21clsfval 22880 . . . 4 (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}))
32fveq1d 6886 . . 3 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆))
43adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆))
5 eqid 2726 . . 3 (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})
6 sseq1 4002 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
76rabbidv 3434 . . . 4 (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
87inteqd 4948 . . 3 (𝑦 = 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
91topopn 22759 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 5337 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 477 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
131topcld 22890 . . . . 5 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
14 sseq2 4003 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1514rspcev 3606 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
1613, 15sylan 579 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
17 intexrab 5333 . . . 4 (∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
1816, 17sylib 217 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
195, 8, 12, 18fvmptd3 7014 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
204, 19eqtrd 2766 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064  {crab 3426  Vcvv 3468  wss 3943  𝒫 cpw 4597   cuni 4902   cint 4943  cmpt 5224  cfv 6536  Topctop 22746  Clsdccld 22871  clsccl 22873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-cld 22874  df-cls 22876
This theorem is referenced by:  cldcls  22897  clscld  22902  clsf  22903  clsval2  22905  clsss  22909  sscls  22911
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