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Theorem clsval 20889
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 20877 . . . 4 (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}))
32fveq1d 6231 . . 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))
51topopn 20759 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
6 elpw2g 4857 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
75, 6syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
87biimpar 501 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
91topcld 20887 . . . . 5 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
10 sseq2 3660 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1110rspcev 3340 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
129, 11sylan 487 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
13 intexrab 4853 . . . 4 (∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
1412, 13sylib 208 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
15 sseq1 3659 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
1615rabbidv 3220 . . . . 5 (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
1716inteqd 4512 . . . 4 (𝑦 = 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
18 eqid 2651 . . . 4 (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})
1917, 18fvmptg 6319 . . 3 ((𝑆 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
208, 14, 19syl2anc 694 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
214, 20eqtrd 2685 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468   cint 4507  cmpt 4762  cfv 5926  Topctop 20746  Clsdccld 20868  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871  df-cls 20873
This theorem is referenced by:  cldcls  20894  clscld  20899  clsf  20900  clsval2  20902  clsss  20906  sscls  20908
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