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Theorem isclo2 22979
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 22978 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3 eleq1w 2811 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
43bibi2d 342 . . . . . . . . . 10 (𝑧 = 𝑤 → ((𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴𝑤𝐴)))
54cbvralvw 3229 . . . . . . . . 9 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴))
65anbi2i 622 . . . . . . . 8 ((∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
7 pm4.24 563 . . . . . . . 8 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
8 raaanv 4517 . . . . . . . 8 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
96, 7, 83bitr4i 303 . . . . . . 7 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)))
10 bibi1 351 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → ((𝑥𝐴𝑤𝐴) ↔ (𝑧𝐴𝑤𝐴)))
1110biimpa 476 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑤𝐴))
1211biimpcd 248 . . . . . . . . . . 11 (𝑧𝐴 → (((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → 𝑤𝐴))
1312ralimdv 3164 . . . . . . . . . 10 (𝑧𝐴 → (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑤𝑦 𝑤𝐴))
1413com12 32 . . . . . . . . 9 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴 → ∀𝑤𝑦 𝑤𝐴))
15 dfss3 3966 . . . . . . . . 9 (𝑦𝐴 ↔ ∀𝑤𝑦 𝑤𝐴)
1614, 15imbitrrdi 251 . . . . . . . 8 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑦𝐴))
1716ralimi 3078 . . . . . . 7 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
189, 17sylbi 216 . . . . . 6 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
19 eleq1w 2811 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
2019imbi1d 341 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑧𝐴𝑦𝐴) ↔ (𝑥𝐴𝑦𝐴)))
2120rspcv 3603 . . . . . . . . 9 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (𝑥𝐴𝑦𝐴)))
22 dfss3 3966 . . . . . . . . . . 11 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2322imbi2i 336 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
24 r19.21v 3174 . . . . . . . . . 10 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
2523, 24bitr4i 278 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))
2621, 25imbitrdi 250 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
27 ssel 3971 . . . . . . . . . . 11 (𝑦𝐴 → (𝑥𝑦𝑥𝐴))
2827com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐴𝑥𝐴))
2928imim2d 57 . . . . . . . . 9 (𝑥𝑦 → ((𝑧𝐴𝑦𝐴) → (𝑧𝐴𝑥𝐴)))
3029ralimdv 3164 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3126, 30jcad 512 . . . . . . 7 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴))))
32 ralbiim 3108 . . . . . . 7 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3331, 32imbitrrdi 251 . . . . . 6 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3418, 33impbid2 225 . . . . 5 (𝑥𝑦 → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3534pm5.32i 574 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3635rexbii 3089 . . 3 (∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3736ralbii 3088 . 2 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
382, 37bitrdi 287 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  cin 3943  wss 3944   cuni 4903  cfv 6542  Topctop 22782  Clsdccld 22907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-topgen 17416  df-top 22783  df-cld 22910
This theorem is referenced by:  connpconn  34781
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