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Theorem isclo 21112
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 3947 . 2 (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
2 isclo.1 . . . . 5 𝑋 = 𝐽
32iscld2 21053 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋𝐴) ∈ 𝐽))
43anbi2d 614 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽)))
5 eltop2 21000 . . . . . 6 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))
6 dfss3 3741 . . . . . . . . . 10 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
7 pm5.501 355 . . . . . . . . . . 11 (𝑥𝐴 → (𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
87ralbidv 3135 . . . . . . . . . 10 (𝑥𝐴 → (∀𝑧𝑦 𝑧𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
96, 8syl5bb 272 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
109anbi2d 614 . . . . . . . 8 (𝑥𝐴 → ((𝑥𝑦𝑦𝐴) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1110rexbidv 3200 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1211ralbiia 3128 . . . . . 6 (∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
135, 12syl6bb 276 . . . . 5 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
14 eltop2 21000 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴))))
15 dfss3 3741 . . . . . . . . . 10 (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 𝑧 ∈ (𝑋𝐴))
16 id 22 . . . . . . . . . . . . . . 15 (𝑧𝑦𝑧𝑦)
17 simpr 471 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → 𝑦𝐽)
18 elunii 4579 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝐽) → 𝑧 𝐽)
1916, 17, 18syl2anr 584 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧 𝐽)
2019, 2syl6eleqr 2861 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧𝑋)
21 eldif 3733 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋𝐴) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝐴))
2221baib 525 . . . . . . . . . . . . 13 (𝑧𝑋 → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
2320, 22syl 17 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
24 eldifn 3884 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋𝐴) → ¬ 𝑥𝐴)
25 nbn2 359 . . . . . . . . . . . . . 14 𝑥𝐴 → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝐴) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2726ad2antrr 705 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2823, 27bitrd 268 . . . . . . . . . . 11 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ (𝑥𝐴𝑧𝐴)))
2928ralbidva 3134 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (∀𝑧𝑦 𝑧 ∈ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3015, 29syl5bb 272 . . . . . . . . 9 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3130anbi2d 614 . . . . . . . 8 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → ((𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3231rexbidva 3197 . . . . . . 7 (𝑥 ∈ (𝑋𝐴) → (∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3332ralbiia 3128 . . . . . 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 616 . . . 4 (𝐽 ∈ Top → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
3635adantr 466 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
37 ralunb 3945 . . . 4 (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
38 simpr 471 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
39 undif 4191 . . . . . 6 (𝐴𝑋 ↔ (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4038, 39sylib 208 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4140raleqdv 3293 . . . 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 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cdif 3720  cun 3721  cin 3722  wss 3723   cuni 4574  cfv 6031  Topctop 20918  Clsdccld 21041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16312  df-top 20919  df-cld 21044
This theorem is referenced by:  isclo2  21113  cvmliftmolem2  31602  cvmlift2lem12  31634
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