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Theorem isclo 21101
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 3995 . 2 (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))
2 isclo.1 . . . . 5 𝑋 = 𝐽
32iscld2 21042 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋𝐴) ∈ 𝐽))
43anbi2d 616 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽)))
5 eltop2 20989 . . . . . 6 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))
6 dfss3 3787 . . . . . . . . . 10 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
7 pm5.501 357 . . . . . . . . . . 11 (𝑥𝐴 → (𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
87ralbidv 3174 . . . . . . . . . 10 (𝑥𝐴 → (∀𝑧𝑦 𝑧𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
96, 8syl5bb 274 . . . . . . . . 9 (𝑥𝐴 → (𝑦𝐴 ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
109anbi2d 616 . . . . . . . 8 (𝑥𝐴 → ((𝑥𝑦𝑦𝐴) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1110rexbidv 3240 . . . . . . 7 (𝑥𝐴 → (∃𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
1211ralbiia 3167 . . . . . 6 (∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
135, 12syl6bb 278 . . . . 5 (𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
14 eltop2 20989 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴))))
15 dfss3 3787 . . . . . . . . . 10 (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 𝑧 ∈ (𝑋𝐴))
16 id 22 . . . . . . . . . . . . . . 15 (𝑧𝑦𝑧𝑦)
17 simpr 473 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → 𝑦𝐽)
18 elunii 4635 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝐽) → 𝑧 𝐽)
1916, 17, 18syl2anr 586 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧 𝐽)
2019, 2syl6eleqr 2896 . . . . . . . . . . . . 13 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → 𝑧𝑋)
21 eldif 3779 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋𝐴) ↔ (𝑧𝑋 ∧ ¬ 𝑧𝐴))
2221baib 527 . . . . . . . . . . . . 13 (𝑧𝑋 → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
2320, 22syl 17 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ ¬ 𝑧𝐴))
24 eldifn 3932 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋𝐴) → ¬ 𝑥𝐴)
25 nbn2 361 . . . . . . . . . . . . . 14 𝑥𝐴 → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2624, 25syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝐴) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2726ad2antrr 708 . . . . . . . . . . . 12 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (¬ 𝑧𝐴 ↔ (𝑥𝐴𝑧𝐴)))
2823, 27bitrd 270 . . . . . . . . . . 11 (((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) ∧ 𝑧𝑦) → (𝑧 ∈ (𝑋𝐴) ↔ (𝑥𝐴𝑧𝐴)))
2928ralbidva 3173 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (∀𝑧𝑦 𝑧 ∈ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3015, 29syl5bb 274 . . . . . . . . 9 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → (𝑦 ⊆ (𝑋𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3130anbi2d 616 . . . . . . . 8 ((𝑥 ∈ (𝑋𝐴) ∧ 𝑦𝐽) → ((𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3231rexbidva 3237 . . . . . . 7 (𝑥 ∈ (𝑋𝐴) → (∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3332ralbiia 3167 . . . . . 6 (∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦𝑦 ⊆ (𝑋𝐴)) ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3414, 33syl6bb 278 . . . . 5 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3513, 34anbi12d 618 . . . 4 (𝐽 ∈ Top → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
3635adantr 468 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽 ∧ (𝑋𝐴) ∈ 𝐽) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))))
37 ralunb 3993 . . . 4 (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
38 simpr 473 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
39 undif 4245 . . . . . 6 (𝐴𝑋 ↔ (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4038, 39sylib 209 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∪ (𝑋𝐴)) = 𝑋)
4140raleqdv 3333 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∀𝑥 ∈ (𝐴 ∪ (𝑋𝐴))∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
4237, 41syl5bbr 276 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((∀𝑥𝐴𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ∧ ∀𝑥 ∈ (𝑋𝐴)∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
434, 36, 423bitrd 296 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴𝐽𝐴 ∈ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
441, 43syl5bb 274 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 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wrex 3097  cdif 3766  cun 3767  cin 3768  wss 3769   cuni 4630  cfv 6097  Topctop 20907  Clsdccld 21030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6060  df-fun 6099  df-fv 6105  df-topgen 16305  df-top 20908  df-cld 21033
This theorem is referenced by:  isclo2  21102  cvmliftmolem2  31582  cvmlift2lem12  31614
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