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Theorem isclo 20801
Description: A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 such that all the points in 𝑦 are in 𝐴 iff 𝑥 is. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
isclo.1 𝑋 = 𝐽
Assertion
Ref Expression
isclo ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem isclo
StepHypRef Expression
1 elin 3774 . 2 (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
2 isclo.1 . . . . 5 𝑋 = 𝐽
32iscld2 20742 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋𝐴) ∈ 𝐽))
43anbi2d 739 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽)))
5 eltop2 20690 . . . . . 6 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))
6 dfss3 3573 . . . . . . . . . 10 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
7 pm5.501 356 . . . . . . . . . . 11 (𝑥𝐴 → (𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
87ralbidv 2980 . . . . . . . . . 10 (𝑥𝐴 → (∀𝑧𝑦 𝑧𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
96, 8syl5bb 272 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
109anbi2d 739 . . . . . . . 8 (𝑥𝐴 → ((𝑥𝑦𝑦𝐴) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1110rexbidv 3045 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1211ralbiia 2973 . . . . . 6 (∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
135, 12syl6bb 276 . . . . 5 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
14 eltop2 20690 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴))))
15 dfss3 3573 . . . . . . . . . 10 (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 𝑧 ∈ (𝑋𝐴))
16 id 22 . . . . . . . . . . . . . . 15 (𝑧𝑦𝑧𝑦)
17 simpr 477 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → 𝑦𝐽)
18 elunii 4407 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝐽) → 𝑧 𝐽)
1916, 17, 18syl2anr 495 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧 𝐽)
2019, 2syl6eleqr 2709 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧𝑋)
21 eldif 3565 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋𝐴) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝐴))
2221baib 943 . . . . . . . . . . . . 13 (𝑧𝑋 → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
2320, 22syl 17 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
24 eldifn 3711 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋𝐴) → ¬ 𝑥𝐴)
25 nbn2 360 . . . . . . . . . . . . . 14 𝑥𝐴 → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝐴) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2726ad2antrr 761 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2823, 27bitrd 268 . . . . . . . . . . 11 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ (𝑥𝐴𝑧𝐴)))
2928ralbidva 2979 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (∀𝑧𝑦 𝑧 ∈ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3015, 29syl5bb 272 . . . . . . . . 9 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3130anbi2d 739 . . . . . . . 8 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → ((𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3231rexbidva 3042 . . . . . . 7 (𝑥 ∈ (𝑋𝐴) → (∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3332ralbiia 2973 . . . . . 6 (∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3414, 33syl6bb 276 . . . . 5 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3513, 34anbi12d 746 . . . 4 (𝐽 ∈ Top → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
3635adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
37 ralunb 3772 . . . 4 (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
38 simpr 477 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
39 undif 4021 . . . . . 6 (𝐴𝑋 ↔ (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4038, 39sylib 208 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4140raleqdv 3133 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
4237, 41syl5bbr 274 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
434, 36, 423bitrd 294 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
441, 43syl5bb 272 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cdif 3552  cun 3553  cin 3554  wss 3555   cuni 4402  cfv 5847  Topctop 20617  Clsdccld 20730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-topgen 16025  df-top 20621  df-cld 20733
This theorem is referenced by:  isclo2  20802  cvmliftmolem2  30972  cvmlift2lem12  31004
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