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Theorem isclo 22573
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 3963 . 2 (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
2 isclo.1 . . . . 5 𝑋 = 𝐽
32iscld2 22514 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋𝐴) ∈ 𝐽))
43anbi2d 630 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽)))
5 eltop2 22460 . . . . . 6 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))
6 dfss3 3969 . . . . . . . . . 10 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
7 pm5.501 367 . . . . . . . . . . 11 (𝑥𝐴 → (𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
87ralbidv 3178 . . . . . . . . . 10 (𝑥𝐴 → (∀𝑧𝑦 𝑧𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
96, 8bitrid 283 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
109anbi2d 630 . . . . . . . 8 (𝑥𝐴 → ((𝑥𝑦𝑦𝐴) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1110rexbidv 3179 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1211ralbiia 3092 . . . . . 6 (∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
135, 12bitrdi 287 . . . . 5 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
14 eltop2 22460 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴))))
15 dfss3 3969 . . . . . . . . . 10 (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 𝑧 ∈ (𝑋𝐴))
16 id 22 . . . . . . . . . . . . . . 15 (𝑧𝑦𝑧𝑦)
17 simpr 486 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → 𝑦𝐽)
18 elunii 4912 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝐽) → 𝑧 𝐽)
1916, 17, 18syl2anr 598 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧 𝐽)
2019, 2eleqtrrdi 2845 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧𝑋)
21 eldif 3957 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋𝐴) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝐴))
2221baib 537 . . . . . . . . . . . . 13 (𝑧𝑋 → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
2320, 22syl 17 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
24 eldifn 4126 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋𝐴) → ¬ 𝑥𝐴)
25 nbn2 371 . . . . . . . . . . . . . 14 𝑥𝐴 → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝐴) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2726ad2antrr 725 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2823, 27bitrd 279 . . . . . . . . . . 11 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ (𝑥𝐴𝑧𝐴)))
2928ralbidva 3176 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (∀𝑧𝑦 𝑧 ∈ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3015, 29bitrid 283 . . . . . . . . 9 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3130anbi2d 630 . . . . . . . 8 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → ((𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3231rexbidva 3177 . . . . . . 7 (𝑥 ∈ (𝑋𝐴) → (∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3332ralbiia 3092 . . . . . 6 (∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3414, 33bitrdi 287 . . . . 5 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3513, 34anbi12d 632 . . . 4 (𝐽 ∈ Top → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
3635adantr 482 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
37 ralunb 4190 . . . 4 (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
38 simpr 486 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
39 undif 4480 . . . . . 6 (𝐴𝑋 ↔ (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4038, 39sylib 217 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4140raleqdv 3326 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
4237, 41bitr3id 285 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
434, 36, 423bitrd 305 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
441, 43bitrid 283 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cdif 3944  cun 3945  cin 3946  wss 3947   cuni 4907  cfv 6540  Topctop 22377  Clsdccld 22502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385  df-top 22378  df-cld 22505
This theorem is referenced by:  isclo2  22574  cvmliftmolem2  34211  cvmlift2lem12  34243
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