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Theorem clsval2 23058
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 3437 . . . . . 6 {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)}
2 clscld.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
32cldopn 23039 . . . . . . . . . . . 12 (𝑧 ∈ (Clsd‘𝐽) → (𝑋𝑧) ∈ 𝐽)
43ad2antrl 728 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝐽)
5 sscon 4143 . . . . . . . . . . . . 13 (𝑆𝑧 → (𝑋𝑧) ⊆ (𝑋𝑆))
65ad2antll 729 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ⊆ (𝑋𝑆))
72topopn 22912 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → 𝑋𝐽)
8 difexg 5329 . . . . . . . . . . . . . 14 (𝑋𝐽 → (𝑋𝑧) ∈ V)
9 elpwg 4603 . . . . . . . . . . . . . 14 ((𝑋𝑧) ∈ V → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
107, 8, 93syl 18 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
1110ad2antrr 726 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
126, 11mpbird 257 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝒫 (𝑋𝑆))
134, 12elind 4200 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
142cldss 23037 . . . . . . . . . . . . 13 (𝑧 ∈ (Clsd‘𝐽) → 𝑧𝑋)
1514ad2antrl 728 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧𝑋)
16 dfss4 4269 . . . . . . . . . . . 12 (𝑧𝑋 ↔ (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1715, 16sylib 218 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1817eqcomd 2743 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧 = (𝑋 ∖ (𝑋𝑧)))
19 difeq2 4120 . . . . . . . . . . 11 (𝑥 = (𝑋𝑧) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑧)))
2019rspceeqv 3645 . . . . . . . . . 10 (((𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2113, 18, 20syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2221ex 412 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
23 simpl 482 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝐽 ∈ Top)
24 elinel1 4201 . . . . . . . . . . . 12 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥𝐽)
252opncld 23041 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
2623, 24, 25syl2an 596 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑋𝑥) ∈ (Clsd‘𝐽))
27 elinel2 4202 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥 ∈ 𝒫 (𝑋𝑆))
2827adantl 481 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ∈ 𝒫 (𝑋𝑆))
2928elpwid 4609 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ⊆ (𝑋𝑆))
3029difss2d 4139 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥𝑋)
31 simplr 769 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆𝑋)
32 ssconb 4142 . . . . . . . . . . . . 13 ((𝑥𝑋𝑆𝑋) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3330, 31, 32syl2anc 584 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3429, 33mpbid 232 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆 ⊆ (𝑋𝑥))
3526, 34jca 511 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)))
36 eleq1 2829 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ (Clsd‘𝐽)))
37 sseq2 4010 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑆𝑧𝑆 ⊆ (𝑋𝑥)))
3836, 37anbi12d 632 . . . . . . . . . 10 (𝑧 = (𝑋𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥))))
3935, 38syl5ibrcom 247 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4039rexlimdva 3155 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4122, 40impbid 212 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
4241abbidv 2808 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
431, 42eqtrid 2789 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
4443inteqd 4951 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
45 difexg 5329 . . . . . . 7 (𝑋𝐽 → (𝑋𝑥) ∈ V)
4645ralrimivw 3150 . . . . . 6 (𝑋𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V)
47 dfiin2g 5032 . . . . . 6 (∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
487, 46, 473syl 18 . . . . 5 (𝐽 ∈ Top → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
4948adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
5044, 49eqtr4d 2780 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
512clsval 23045 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧})
52 uniiun 5058 . . . . . 6 (𝐽 ∩ 𝒫 (𝑋𝑆)) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥
5352difeq2i 4123 . . . . 5 (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥)
5453a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
55 0opn 22910 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5655adantr 480 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝐽)
57 0elpw 5356 . . . . . . 7 ∅ ∈ 𝒫 (𝑋𝑆)
5857a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝒫 (𝑋𝑆))
5956, 58elind 4200 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
60 ne0i 4341 . . . . 5 (∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → (𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅)
61 iindif2 5077 . . . . 5 ((𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅ → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
6259, 60, 613syl 18 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
6354, 62eqtr4d 2780 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
6450, 51, 633eqtr4d 2787 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
65 difssd 4137 . . . 4 (𝑆𝑋 → (𝑋𝑆) ⊆ 𝑋)
662ntrval 23044 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6765, 66sylan2 593 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6867difeq2d 4126 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
6964, 68eqtr4d 2780 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907   cint 4946   ciun 4991   ciin 4992  cfv 6561  Topctop 22899  Clsdccld 23024  intcnt 23025  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029
This theorem is referenced by:  ntrval2  23059  clsdif  23061  cmclsopn  23070  bcth3  25365
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