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Theorem clsval2 21134
Description: Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsval2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))

Proof of Theorem clsval2
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 3064 . . . . . 6 {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)}
2 clscld.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
32cldopn 21115 . . . . . . . . . . . 12 (𝑧 ∈ (Clsd‘𝐽) → (𝑋𝑧) ∈ 𝐽)
43ad2antrl 719 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝐽)
5 sscon 3906 . . . . . . . . . . . . 13 (𝑆𝑧 → (𝑋𝑧) ⊆ (𝑋𝑆))
65ad2antll 720 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ⊆ (𝑋𝑆))
72topopn 20990 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → 𝑋𝐽)
8 difexg 4969 . . . . . . . . . . . . . 14 (𝑋𝐽 → (𝑋𝑧) ∈ V)
9 elpwg 4323 . . . . . . . . . . . . . 14 ((𝑋𝑧) ∈ V → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
107, 8, 93syl 18 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
1110ad2antrr 717 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
126, 11mpbird 248 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝒫 (𝑋𝑆))
134, 12elind 3960 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
142cldss 21113 . . . . . . . . . . . . 13 (𝑧 ∈ (Clsd‘𝐽) → 𝑧𝑋)
1514ad2antrl 719 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧𝑋)
16 dfss4 4023 . . . . . . . . . . . 12 (𝑧𝑋 ↔ (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1715, 16sylib 209 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1817eqcomd 2771 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧 = (𝑋 ∖ (𝑋𝑧)))
19 difeq2 3884 . . . . . . . . . . 11 (𝑥 = (𝑋𝑧) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑧)))
2019rspceeqv 3479 . . . . . . . . . 10 (((𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2113, 18, 20syl2anc 579 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2221ex 401 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
23 simpl 474 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝐽 ∈ Top)
24 elin 3958 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) ↔ (𝑥𝐽𝑥 ∈ 𝒫 (𝑋𝑆)))
2524simplbi 491 . . . . . . . . . . . 12 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥𝐽)
262opncld 21117 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
2723, 25, 26syl2an 589 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑋𝑥) ∈ (Clsd‘𝐽))
2824simprbi 490 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥 ∈ 𝒫 (𝑋𝑆))
2928adantl 473 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ∈ 𝒫 (𝑋𝑆))
3029elpwid 4327 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ⊆ (𝑋𝑆))
3130difss2d 3902 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥𝑋)
32 simplr 785 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆𝑋)
33 ssconb 3905 . . . . . . . . . . . . 13 ((𝑥𝑋𝑆𝑋) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3431, 32, 33syl2anc 579 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3530, 34mpbid 223 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆 ⊆ (𝑋𝑥))
3627, 35jca 507 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)))
37 eleq1 2832 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ (Clsd‘𝐽)))
38 sseq2 3787 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑆𝑧𝑆 ⊆ (𝑋𝑥)))
3937, 38anbi12d 624 . . . . . . . . . 10 (𝑧 = (𝑋𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥))))
4036, 39syl5ibrcom 238 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4140rexlimdva 3178 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4222, 41impbid 203 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
4342abbidv 2884 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
441, 43syl5eq 2811 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
4544inteqd 4638 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
46 difexg 4969 . . . . . . 7 (𝑋𝐽 → (𝑋𝑥) ∈ V)
4746ralrimivw 3114 . . . . . 6 (𝑋𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V)
48 dfiin2g 4709 . . . . . 6 (∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
497, 47, 483syl 18 . . . . 5 (𝐽 ∈ Top → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
5049adantr 472 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
5145, 50eqtr4d 2802 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
522clsval 21121 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧})
53 uniiun 4729 . . . . . 6 (𝐽 ∩ 𝒫 (𝑋𝑆)) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥
5453difeq2i 3887 . . . . 5 (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥)
5554a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
56 0opn 20988 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5756adantr 472 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝐽)
58 0elpw 4992 . . . . . . 7 ∅ ∈ 𝒫 (𝑋𝑆)
5958a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝒫 (𝑋𝑆))
6057, 59elind 3960 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
61 ne0i 4085 . . . . 5 (∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → (𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅)
62 iindif2 4745 . . . . 5 ((𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅ → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
6360, 61, 623syl 18 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
6455, 63eqtr4d 2802 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
6551, 52, 643eqtr4d 2809 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
66 difssd 3900 . . . 4 (𝑆𝑋 → (𝑋𝑆) ⊆ 𝑋)
672ntrval 21120 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6866, 67sylan2 586 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6968difeq2d 3890 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
7065, 69eqtr4d 2802 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315   cuni 4594   cint 4633   ciun 4676   ciin 4677  cfv 6068  Topctop 20977  Clsdccld 21100  intcnt 21101  clsccl 21102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-top 20978  df-cld 21103  df-ntr 21104  df-cls 21105
This theorem is referenced by:  ntrval2  21135  clsdif  21137  cmclsopn  21146  bcth3  23408
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