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Theorem clsval 21219
 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 21207 . . . 4 (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}))
32fveq1d 6439 . . 3 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆))
43adantr 474 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆))
5 eqid 2825 . . 3 (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})
6 sseq1 3851 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
76rabbidv 3402 . . . 4 (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
87inteqd 4704 . . 3 (𝑦 = 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
91topopn 21088 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 5051 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 471 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
131topcld 21217 . . . . 5 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
14 sseq2 3852 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1514rspcev 3526 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
1613, 15sylan 575 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
17 intexrab 5047 . . . 4 (∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
1816, 17sylib 210 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
195, 8, 12, 18fvmptd3 6555 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
204, 19eqtrd 2861 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∃wrex 3118  {crab 3121  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4380  ∪ cuni 4660  ∩ cint 4699   ↦ cmpt 4954  ‘cfv 6127  Topctop 21075  Clsdccld 21198  clsccl 21200 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-top 21076  df-cld 21201  df-cls 21203 This theorem is referenced by:  cldcls  21224  clscld  21229  clsf  21230  clsval2  21232  clsss  21236  sscls  21238
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