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Theorem isclo2 20797
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 20796 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))
3 eleq1 2692 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
43bibi2d 332 . . . . . . . . . 10 (𝑧 = 𝑤 → ((𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴𝑤𝐴)))
54cbvralv 3164 . . . . . . . . 9 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴))
65anbi2i 729 . . . . . . . 8 ((∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
7 pm4.24 674 . . . . . . . 8 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
8 raaanv 4060 . . . . . . . 8 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑤𝑦 (𝑥𝐴𝑤𝐴)))
96, 7, 83bitr4i 292 . . . . . . 7 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)))
10 bibi1 341 . . . . . . . . . . . . 13 ((𝑥𝐴𝑧𝐴) → ((𝑥𝐴𝑤𝐴) ↔ (𝑧𝐴𝑤𝐴)))
1110biimpa 501 . . . . . . . . . . . 12 (((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑤𝐴))
1211biimpcd 239 . . . . . . . . . . 11 (𝑧𝐴 → (((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → 𝑤𝐴))
1312ralimdv 2962 . . . . . . . . . 10 (𝑧𝐴 → (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑤𝑦 𝑤𝐴))
1413com12 32 . . . . . . . . 9 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴 → ∀𝑤𝑦 𝑤𝐴))
15 dfss3 3578 . . . . . . . . 9 (𝑦𝐴 ↔ ∀𝑤𝑦 𝑤𝐴)
1614, 15syl6ibr 242 . . . . . . . 8 (∀𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → (𝑧𝐴𝑦𝐴))
1716ralimi 2952 . . . . . . 7 (∀𝑧𝑦𝑤𝑦 ((𝑥𝐴𝑧𝐴) ∧ (𝑥𝐴𝑤𝐴)) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
189, 17sylbi 207 . . . . . 6 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))
19 eleq1 2692 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
2019imbi1d 331 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑧𝐴𝑦𝐴) ↔ (𝑥𝐴𝑦𝐴)))
2120rspcv 3296 . . . . . . . . 9 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (𝑥𝐴𝑦𝐴)))
22 dfss3 3578 . . . . . . . . . . 11 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2322imbi2i 326 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
24 r19.21v 2959 . . . . . . . . . 10 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (𝑥𝐴 → ∀𝑧𝑦 𝑧𝐴))
2523, 24bitr4i 267 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) ↔ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))
2621, 25syl6ib 241 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
27 ssel 3582 . . . . . . . . . . 11 (𝑦𝐴 → (𝑥𝑦𝑥𝐴))
2827com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐴𝑥𝐴))
2928imim2d 57 . . . . . . . . 9 (𝑥𝑦 → ((𝑧𝐴𝑦𝐴) → (𝑧𝐴𝑥𝐴)))
3029ralimdv 2962 . . . . . . . 8 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3126, 30jcad 555 . . . . . . 7 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴))))
32 ralbiim 3067 . . . . . . 7 (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ∧ ∀𝑧𝑦 (𝑧𝐴𝑥𝐴)))
3331, 32syl6ibr 242 . . . . . 6 (𝑥𝑦 → (∀𝑧𝑦 (𝑧𝐴𝑦𝐴) → ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)))
3418, 33impbid2 216 . . . . 5 (𝑥𝑦 → (∀𝑧𝑦 (𝑥𝐴𝑧𝐴) ↔ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3534pm5.32i 668 . . . 4 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3635rexbii 3039 . . 3 (∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∃𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
3736ralbii 2979 . 2 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴)))
382, 37syl6bb 276 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cin 3559  wss 3560   cuni 4407  cfv 5850  Topctop 20612  Clsdccld 20725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-topgen 16020  df-top 20616  df-cld 20728
This theorem is referenced by:  connpconn  30917
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