MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clsval2 Structured version   Visualization version   GIF version

Theorem clsval2 20902
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 2950 . . . . . 6 {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)}
2 clscld.1 . . . . . . . . . . . . 13 𝑋 = 𝐽
32cldopn 20883 . . . . . . . . . . . 12 (𝑧 ∈ (Clsd‘𝐽) → (𝑋𝑧) ∈ 𝐽)
43ad2antrl 764 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝐽)
5 sscon 3777 . . . . . . . . . . . . 13 (𝑆𝑧 → (𝑋𝑧) ⊆ (𝑋𝑆))
65ad2antll 765 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ⊆ (𝑋𝑆))
72topopn 20759 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → 𝑋𝐽)
8 difexg 4841 . . . . . . . . . . . . . 14 (𝑋𝐽 → (𝑋𝑧) ∈ V)
9 elpwg 4199 . . . . . . . . . . . . . 14 ((𝑋𝑧) ∈ V → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
107, 8, 93syl 18 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
1110ad2antrr 762 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → ((𝑋𝑧) ∈ 𝒫 (𝑋𝑆) ↔ (𝑋𝑧) ⊆ (𝑋𝑆)))
126, 11mpbird 247 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ 𝒫 (𝑋𝑆))
134, 12elind 3831 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
142cldss 20881 . . . . . . . . . . . . 13 (𝑧 ∈ (Clsd‘𝐽) → 𝑧𝑋)
1514ad2antrl 764 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧𝑋)
16 dfss4 3891 . . . . . . . . . . . 12 (𝑧𝑋 ↔ (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1715, 16sylib 208 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → (𝑋 ∖ (𝑋𝑧)) = 𝑧)
1817eqcomd 2657 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → 𝑧 = (𝑋 ∖ (𝑋𝑧)))
19 difeq2 3755 . . . . . . . . . . . 12 (𝑥 = (𝑋𝑧) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑧)))
2019eqeq2d 2661 . . . . . . . . . . 11 (𝑥 = (𝑋𝑧) → (𝑧 = (𝑋𝑥) ↔ 𝑧 = (𝑋 ∖ (𝑋𝑧))))
2120rspcev 3340 . . . . . . . . . 10 (((𝑋𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2213, 18, 21syl2anc 694 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥))
2322ex 449 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
24 simpl 472 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝐽 ∈ Top)
25 elin 3829 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) ↔ (𝑥𝐽𝑥 ∈ 𝒫 (𝑋𝑆)))
2625simplbi 475 . . . . . . . . . . . 12 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥𝐽)
272opncld 20885 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (𝑋𝑥) ∈ (Clsd‘𝐽))
2824, 26, 27syl2an 493 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑋𝑥) ∈ (Clsd‘𝐽))
2925simprbi 479 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → 𝑥 ∈ 𝒫 (𝑋𝑆))
3029adantl 481 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ∈ 𝒫 (𝑋𝑆))
3130elpwid 4203 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥 ⊆ (𝑋𝑆))
3231difss2d 3773 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑥𝑋)
33 simplr 807 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆𝑋)
34 ssconb 3776 . . . . . . . . . . . . 13 ((𝑥𝑋𝑆𝑋) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3532, 33, 34syl2anc 694 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑥 ⊆ (𝑋𝑆) ↔ 𝑆 ⊆ (𝑋𝑥)))
3631, 35mpbid 222 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → 𝑆 ⊆ (𝑋𝑥))
3728, 36jca 553 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥)))
38 eleq1 2718 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ (Clsd‘𝐽)))
39 sseq2 3660 . . . . . . . . . . 11 (𝑧 = (𝑋𝑥) → (𝑆𝑧𝑆 ⊆ (𝑋𝑥)))
4038, 39anbi12d 747 . . . . . . . . . 10 (𝑧 = (𝑋𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ((𝑋𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋𝑥))))
4137, 40syl5ibrcom 237 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))) → (𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4241rexlimdva 3060 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)))
4323, 42impbid 202 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)))
4443abbidv 2770 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
451, 44syl5eq 2697 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
4645inteqd 4512 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
47 difexg 4841 . . . . . . 7 (𝑋𝐽 → (𝑋𝑥) ∈ V)
4847ralrimivw 2996 . . . . . 6 (𝑋𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V)
49 dfiin2g 4585 . . . . . 6 (∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) ∈ V → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
507, 48, 493syl 18 . . . . 5 (𝐽 ∈ Top → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
5150adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑧 = (𝑋𝑥)})
5246, 51eqtr4d 2688 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧} = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
532clsval 20889 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆𝑧})
54 uniiun 4605 . . . . . 6 (𝐽 ∩ 𝒫 (𝑋𝑆)) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥
5554difeq2i 3758 . . . . 5 (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥)
5655a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
57 0opn 20757 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5857adantr 480 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝐽)
59 0elpw 4864 . . . . . . 7 ∅ ∈ 𝒫 (𝑋𝑆)
6059a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ 𝒫 (𝑋𝑆))
6158, 60elind 3831 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)))
62 ne0i 3954 . . . . 5 (∅ ∈ (𝐽 ∩ 𝒫 (𝑋𝑆)) → (𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅)
63 iindif2 4621 . . . . 5 ((𝐽 ∩ 𝒫 (𝑋𝑆)) ≠ ∅ → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
6461, 62, 633syl 18 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥) = (𝑋 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))𝑥))
6556, 64eqtr4d 2688 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))) = 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋𝑆))(𝑋𝑥))
6652, 53, 653eqtr4d 2695 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
67 difssd 3771 . . . 4 (𝑆𝑋 → (𝑋𝑆) ⊆ 𝑋)
682ntrval 20888 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
6967, 68sylan2 490 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋𝑆)) = (𝐽 ∩ 𝒫 (𝑋𝑆)))
7069difeq2d 3761 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))) = (𝑋 (𝐽 ∩ 𝒫 (𝑋𝑆))))
7166, 70eqtr4d 2688 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507   ciun 4552   ciin 4553  cfv 5926  Topctop 20746  Clsdccld 20868  intcnt 20869  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by:  ntrval2  20903  clsdif  20905  cmclsopn  20914  bcth3  23174
  Copyright terms: Public domain W3C validator