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Theorem ismntop 30379
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 30378 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
2 haustop 21337 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
32adantl 473 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → 𝐽 ∈ Top)
43biantrurd 530 . . . . . . 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 21789 . . . . . . . . . . . . 13 ≃ Er Top
6 errel 7920 . . . . . . . . . . . . 13 ( ≃ Er Top → Rel ≃ )
7 relelec 7954 . . . . . . . . . . . . 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 21791 . . . . . . . . . . . 12 ((TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢) ↔ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))
108, 9bitr2i 265 . . . . . . . . . . 11 ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110a1i 11 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))
1211anbi2d 742 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1312rexbidv 3190 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
14132ralbidv 3127 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
15 islly 21473 . . . . . . . 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 301 . . . . . 6 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))
1817pm5.32da 676 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
1918anbi2d 742 . . . 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 1081 . . . 4 ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
21 3anass 1081 . . . 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 303 . . 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 472 . 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 268 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 196  wa 383  w3a 1072  wcel 2139  wral 3050  wrex 3051  cin 3714  𝒫 cpw 4302   class class class wbr 4804  Rel wrel 5271  cfv 6049  (class class class)co 6813   Er wer 7908  [cec 7909  0cn0 11484  t crest 16283  TopOpenctopn 16284  Topctop 20900  Hauscha 21314  2nd𝜔c2ndc 21443  Locally clly 21469  chmph 21759  𝔼hilcehl 23372  ManTopcmntop 30375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-1o 7729  df-er 7911  df-ec 7913  df-map 8025  df-top 20901  df-topon 20918  df-cn 21233  df-haus 21321  df-lly 21471  df-hmeo 21760  df-hmph 21761  df-mntop 30376
This theorem is referenced by: (None)
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