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Theorem ismntop 34027
Description: Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)
Assertion
Ref Expression
ismntop ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
Distinct variable groups:   𝑢,𝐽,𝑥,𝑦   𝑢,𝑁,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑢)

Proof of Theorem ismntop
StepHypRef Expression
1 ismntoplly 34026 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
2 haustop 23339 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
32adantl 481 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → 𝐽 ∈ Top)
43biantrurd 532 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
5 hmpher 23792 . . . . . . . . . . . . 13 ≃ Er Top
6 errel 8754 . . . . . . . . . . . . 13 ( ≃ Er Top → Rel ≃ )
7 relelec 8792 . . . . . . . . . . . . 13 (Rel ≃ → ((𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢)))
85, 6, 7mp2b 10 . . . . . . . . . . . 12 ((𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢))
9 hmphsymb 23794 . . . . . . . . . . . 12 ((TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢) ↔ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))
108, 9bitr2i 276 . . . . . . . . . . 11 ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110a1i 11 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))
1211anbi2d 630 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1312rexbidv 3179 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
14132ralbidv 3221 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
15 islly 23476 . . . . . . . 8 (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1615a1i 11 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
174, 14, 163bitr4rd 312 . . . . . 6 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))
1817pm5.32da 579 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
1918anbi2d 630 . . . 4 (𝑁 ∈ ℕ0 → ((𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))))
20 3anass 1095 . . . 4 ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
21 3anass 1095 . . . 4 ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
2219, 20, 213bitr4g 314 . . 3 (𝑁 ∈ ℕ0 → ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
2322adantr 480 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
241, 23bitrd 279 1 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wcel 2108  wral 3061  wrex 3070  cin 3950  𝒫 cpw 4600   class class class wbr 5143  Rel wrel 5690  cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743  0cn0 12526  t crest 17465  TopOpenctopn 17466  Topctop 22899  Hauscha 23316  2ndωc2ndc 23446  Locally clly 23472  chmph 23762  𝔼hilcehl 25418  ManTopcmntop 34023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-ec 8747  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235  df-haus 23323  df-lly 23474  df-hmeo 23763  df-hmph 23764  df-mntop 34024
This theorem is referenced by: (None)
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