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Theorem isclo2 21985
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 21984 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3 eleq1w 2820 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
43bibi2d 346 . . . . . . . . . 10 (𝑧 = 𝑤 → ((𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴𝑤𝐴)))
54cbvralvw 3358 . . . . . . . . 9 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴))
65anbi2i 626 . . . . . . . 8 ((∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
7 pm4.24 567 . . . . . . . 8 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
8 raaanv 4433 . . . . . . . 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 3101 . . . . . . . . . 10 (𝑧𝐴 → (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑤𝑦 𝑤𝐴))
1413com12 32 . . . . . . . . 9 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴 → ∀𝑤𝑦 𝑤𝐴))
15 dfss3 3888 . . . . . . . . 9 (𝑦𝐴 ↔ ∀𝑤𝑦 𝑤𝐴)
1614, 15syl6ibr 255 . . . . . . . 8 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑦𝐴))
1716ralimi 3083 . . . . . . 7 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
189, 17sylbi 220 . . . . . 6 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
19 eleq1w 2820 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
2019imbi1d 345 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑧𝐴𝑦𝐴) ↔ (𝑥𝐴𝑦𝐴)))
2120rspcv 3532 . . . . . . . . 9 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (𝑥𝐴𝑦𝐴)))
22 dfss3 3888 . . . . . . . . . . 11 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2322imbi2i 339 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
24 r19.21v 3098 . . . . . . . . . 10 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
2523, 24bitr4i 281 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))
2621, 25syl6ib 254 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
27 ssel 3893 . . . . . . . . . . 11 (𝑦𝐴 → (𝑥𝑦𝑥𝐴))
2827com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐴𝑥𝐴))
2928imim2d 57 . . . . . . . . 9 (𝑥𝑦 → ((𝑧𝐴𝑦𝐴) → (𝑧𝐴𝑥𝐴)))
3029ralimdv 3101 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3126, 30jcad 516 . . . . . . 7 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴))))
32 ralbiim 3096 . . . . . . 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 3170 . . 3 (∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3736ralbii 3088 . 2 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
382, 37bitrdi 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 1543  wcel 2110  wral 3061  wrex 3062  cin 3865  wss 3866   cuni 4819  cfv 6380  Topctop 21790  Clsdccld 21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-topgen 16948  df-top 21791  df-cld 21916
This theorem is referenced by:  connpconn  32910
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