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Theorem isclo2 21703
 Description: A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 of 𝑥 which is either disjoint from 𝐴 or contained in 𝐴. (Contributed by Mario Carneiro, 7-Jul-2015.)
Hypothesis
Ref Expression
isclo.1 𝑋 = 𝐽
Assertion
Ref Expression
isclo2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem isclo2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 isclo.1 . . 3 𝑋 = 𝐽
21isclo 21702 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3 eleq1w 2872 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
43bibi2d 346 . . . . . . . . . 10 (𝑧 = 𝑤 → ((𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴𝑤𝐴)))
54cbvralvw 3396 . . . . . . . . 9 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴))
65anbi2i 625 . . . . . . . 8 ((∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
7 pm4.24 567 . . . . . . . 8 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
8 raaanv 4419 . . . . . . . 8 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
96, 7, 83bitr4i 306 . . . . . . 7 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)))
10 bibi1 355 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → ((𝑥𝐴𝑤𝐴) ↔ (𝑧𝐴𝑤𝐴)))
1110biimpa 480 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑤𝐴))
1211biimpcd 252 . . . . . . . . . . 11 (𝑧𝐴 → (((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → 𝑤𝐴))
1312ralimdv 3145 . . . . . . . . . 10 (𝑧𝐴 → (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑤𝑦 𝑤𝐴))
1413com12 32 . . . . . . . . 9 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴 → ∀𝑤𝑦 𝑤𝐴))
15 dfss3 3903 . . . . . . . . 9 (𝑦𝐴 ↔ ∀𝑤𝑦 𝑤𝐴)
1614, 15syl6ibr 255 . . . . . . . 8 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑦𝐴))
1716ralimi 3128 . . . . . . 7 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
189, 17sylbi 220 . . . . . 6 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
19 eleq1w 2872 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
2019imbi1d 345 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑧𝐴𝑦𝐴) ↔ (𝑥𝐴𝑦𝐴)))
2120rspcv 3566 . . . . . . . . 9 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (𝑥𝐴𝑦𝐴)))
22 dfss3 3903 . . . . . . . . . . 11 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2322imbi2i 339 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
24 r19.21v 3142 . . . . . . . . . 10 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
2523, 24bitr4i 281 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))
2621, 25syl6ib 254 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
27 ssel 3908 . . . . . . . . . . 11 (𝑦𝐴 → (𝑥𝑦𝑥𝐴))
2827com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐴𝑥𝐴))
2928imim2d 57 . . . . . . . . 9 (𝑥𝑦 → ((𝑧𝐴𝑦𝐴) → (𝑧𝐴𝑥𝐴)))
3029ralimdv 3145 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3126, 30jcad 516 . . . . . . 7 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴))))
32 ralbiim 3141 . . . . . . 7 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3331, 32syl6ibr 255 . . . . . 6 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3418, 33impbid2 229 . . . . 5 (𝑥𝑦 → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3534pm5.32i 578 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3635rexbii 3210 . . 3 (∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3736ralbii 3133 . 2 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
382, 37syl6bb 290 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  ∪ cuni 4801  ‘cfv 6325  Topctop 21508  Clsdccld 21631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-topgen 16712  df-top 21509  df-cld 21634 This theorem is referenced by:  connpconn  32610
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