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Theorem ismntop 31499
 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 31498 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
2 haustop 22036 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
32adantl 485 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → 𝐽 ∈ Top)
43biantrurd 536 . . . . . . 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 22489 . . . . . . . . . . . . 13 ≃ Er Top
6 errel 8313 . . . . . . . . . . . . 13 ( ≃ Er Top → Rel ≃ )
7 relelec 8349 . . . . . . . . . . . . 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 22491 . . . . . . . . . . . 12 ((TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢) ↔ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))
108, 9bitr2i 279 . . . . . . . . . . 11 ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110a1i 11 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))
1211anbi2d 631 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1312rexbidv 3221 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
14132ralbidv 3128 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
15 islly 22173 . . . . . . . 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 315 . . . . . 6 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))
1817pm5.32da 582 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
1918anbi2d 631 . . . 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 1092 . . . 4 ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
21 3anass 1092 . . . 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 317 . . 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 484 . 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 282 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 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ∩ cin 3859  𝒫 cpw 4497   class class class wbr 5035  Rel wrel 5532  ‘cfv 6339  (class class class)co 7155   Er wer 8301  [cec 8302  ℕ0cn0 11939   ↾t crest 16757  TopOpenctopn 16758  Topctop 21598  Hauscha 22013  2ndωc2ndc 22143  Locally clly 22169   ≃ chmph 22459  𝔼hilcehl 24089  ManTopcmntop 31495 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 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-1o 8117  df-er 8304  df-ec 8306  df-map 8423  df-top 21599  df-topon 21616  df-cn 21932  df-haus 22020  df-lly 22171  df-hmeo 22460  df-hmph 22461  df-mntop 31496 This theorem is referenced by: (None)
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