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Theorem clsval2 22874
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 3432 . . . . . 6 {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)}
2 clscld.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
32cldopn 22855 . . . . . . . . . . . 12 (𝑧 ∈ (Clsd‘𝐽) → (𝑋𝑧) ∈ 𝐽)
43ad2antrl 725 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝐽)
5 sscon 4138 . . . . . . . . . . . . 13 (𝑆𝑧 → (𝑋𝑧) ⊆ (𝑋𝑆))
65ad2antll 726 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ⊆ (𝑋𝑆))
72topopn 22728 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → 𝑋𝐽)
8 difexg 5327 . . . . . . . . . . . . . 14 (𝑋𝐽 → (𝑋𝑧) ∈ V)
9 elpwg 4605 . . . . . . . . . . . . . 14 ((𝑋𝑧) ∈ V → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
107, 8, 93syl 18 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
1110ad2antrr 723 . . . . . . . . . . . 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 4194 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
142cldss 22853 . . . . . . . . . . . . 13 (𝑧 ∈ (Clsd‘𝐽) → 𝑧𝑋)
1514ad2antrl 725 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧𝑋)
16 dfss4 4258 . . . . . . . . . . . 12 (𝑧𝑋 ↔ (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1715, 16sylib 217 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1817eqcomd 2737 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧 = (𝑋 ∖ (𝑋𝑧)))
19 difeq2 4116 . . . . . . . . . . 11 (𝑥 = (𝑋𝑧) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑧)))
2019rspceeqv 3633 . . . . . . . . . 10 (((𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2113, 18, 20syl2anc 583 . . . . . . . . 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 4195 . . . . . . . . . . . 12 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥𝐽)
252opncld 22857 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
2623, 24, 25syl2an 595 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑋𝑥) ∈ (Clsd‘𝐽))
27 elinel2 4196 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥 ∈ 𝒫 (𝑋𝑆))
2827adantl 481 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ∈ 𝒫 (𝑋𝑆))
2928elpwid 4611 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ⊆ (𝑋𝑆))
3029difss2d 4134 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥𝑋)
31 simplr 766 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆𝑋)
32 ssconb 4137 . . . . . . . . . . . . 13 ((𝑥𝑋𝑆𝑋) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3330, 31, 32syl2anc 583 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3429, 33mpbid 231 . . . . . . . . . . 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 2820 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ (Clsd‘𝐽)))
37 sseq2 4008 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑆𝑧𝑆 ⊆ (𝑋𝑥)))
3836, 37anbi12d 630 . . . . . . . . . 10 (𝑧 = (𝑋𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥))))
3935, 38syl5ibrcom 246 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4039rexlimdva 3154 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4122, 40impbid 211 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
4241abbidv 2800 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
431, 42eqtrid 2783 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
4443inteqd 4955 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
45 difexg 5327 . . . . . . 7 (𝑋𝐽 → (𝑋𝑥) ∈ V)
4645ralrimivw 3149 . . . . . 6 (𝑋𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V)
47 dfiin2g 5035 . . . . . 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 2774 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
512clsval 22861 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧})
52 uniiun 5061 . . . . . 6 (𝐽 ∩ 𝒫 (𝑋𝑆)) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥
5352difeq2i 4119 . . . . 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 22726 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5655adantr 480 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝐽)
57 0elpw 5354 . . . . . . 7 ∅ ∈ 𝒫 (𝑋𝑆)
5857a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝒫 (𝑋𝑆))
5956, 58elind 4194 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
60 ne0i 4334 . . . . 5 (∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → (𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅)
61 iindif2 5080 . . . . 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 2774 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
6450, 51, 633eqtr4d 2781 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
65 difssd 4132 . . . 4 (𝑆𝑋 → (𝑋𝑆) ⊆ 𝑋)
662ntrval 22860 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6765, 66sylan2 592 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6867difeq2d 4122 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
6964, 68eqtr4d 2774 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wne 2939  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  cdif 3945  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602   cuni 4908   cint 4950   ciun 4997   ciin 4998  cfv 6543  Topctop 22715  Clsdccld 22840  intcnt 22841  clsccl 22842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22716  df-cld 22843  df-ntr 22844  df-cls 22845
This theorem is referenced by:  ntrval2  22875  clsdif  22877  cmclsopn  22886  bcth3  25179
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