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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 ∧ 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 20577 . . . 4 (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}))
32fveq1d 6086 . . 3 (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆))
43adantr 479 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆))
51topopn 20474 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
6 elpw2g 4745 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
75, 6syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
87biimpar 500 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
91topcld 20587 . . . . 5 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
10 sseq2 3585 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1110rspcev 3277 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
129, 11sylan 486 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
13 intexrab 4741 . . . 4 (∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
1412, 13sylib 206 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V)
15 sseq1 3584 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
1615rabbidv 3159 . . . . 5 (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
1716inteqd 4405 . . . 4 (𝑦 = 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
18 eqid 2605 . . . 4 (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})
1917, 18fvmptg 6170 . . 3 ((𝑆 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ V) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
208, 14, 19syl2anc 690 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
214, 20eqtrd 2639 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
Colors of variables: wff setvar class
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
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