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Theorem clsval 22959
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 22947 . . . 4 (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}))
32fveq1d 6902 . . 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))
5 eqid 2727 . . 3 (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥}) = (𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})
6 sseq1 4005 . . . . 5 (𝑦 = 𝑆 → (𝑦𝑥𝑆𝑥))
76rabbidv 3436 . . . 4 (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
87inteqd 4956 . . 3 (𝑦 = 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
91topopn 22826 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 5348 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 476 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
131topcld 22957 . . . . 5 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
14 sseq2 4006 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
1514rspcev 3609 . . . . 5 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
1613, 15sylan 578 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆𝑥)
17 intexrab 5344 . . . 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 7031 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑦 ∈ 𝒫 𝑋 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦𝑥})‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
204, 19eqtrd 2767 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3066  {crab 3428  Vcvv 3471  wss 3947  𝒫 cpw 4604   cuni 4910   cint 4951  cmpt 5233  cfv 6551  Topctop 22813  Clsdccld 22938  clsccl 22940
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-top 22814  df-cld 22941  df-cls 22943
This theorem is referenced by:  cldcls  22964  clscld  22969  clsf  22970  clsval2  22972  clsss  22976  sscls  22978
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