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Theorem ismntop 29398
 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 29397 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
2 haustop 20945 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
32adantl 481 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → 𝐽 ∈ Top)
43biantrurd 528 . . . . . . 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 21397 . . . . . . . . . . . . 13 ≃ Er Top
6 errel 7638 . . . . . . . . . . . . 13 ( ≃ Er Top → Rel ≃ )
7 relelec 7674 . . . . . . . . . . . . 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 21399 . . . . . . . . . . . 12 ((TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢) ↔ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))
108, 9bitr2i 264 . . . . . . . . . . 11 ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110a1i 11 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))
1211anbi2d 736 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1312rexbidv 3034 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
14132ralbidv 2972 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
15 islly 21081 . . . . . . . 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 300 . . . . . 6 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))
1817pm5.32da 671 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
1918anbi2d 736 . . . 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 1035 . . . 4 ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
21 3anass 1035 . . . 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 302 . . 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 267 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 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  𝒫 cpw 4108   class class class wbr 4583  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549   Er wer 7626  [cec 7627  ℕ0cn0 11169   ↾t crest 15904  TopOpenctopn 15905  Topctop 20517  Hauscha 20922  2nd𝜔c2ndc 21051  Locally clly 21077   ≃ chmph 21367  𝔼hilcehl 22980  ManTopcmntop 29394 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-ec 7631  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-haus 20929  df-lly 21079  df-hmeo 21368  df-hmph 21369  df-mntop 29395 This theorem is referenced by: (None)
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