Theorem ismntop 34210

Description: Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)

Assertion

Ref	Expression
ismntop	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))

Distinct variable groups:   𝑢,𝐽,𝑥,𝑦   𝑢,𝑁,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑢)

Proof of Theorem ismntop

Step	Hyp	Ref	Expression
1		ismntoplly 34209	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
2		haustop 23314	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 482	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 537	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
5		hmpher 23767	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8643	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8681	. . . . . . . . . . . . 13 ⊢ (Rel ≃ → ((𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ ↔ (TopOpen‘(𝔼hil‘𝑁)) ≃ (𝐽 ↾t 𝑢)))
8	5, 6, 7	mp2b 10	. . . . . . . . . . . 12 ⊢ ((𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ ↔ (TopOpen‘(𝔼hil‘𝑁)) ≃ (𝐽 ↾t 𝑢))
9		hmphsymb 23769	. . . . . . . . . . . 12 ⊢ ((TopOpen‘(𝔼hil‘𝑁)) ≃ (𝐽 ↾t 𝑢) ↔ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)))
10	8, 9	bitr2i 277	. . . . . . . . . . 11 ⊢ ((𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)) ↔ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → ((𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)) ↔ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))
12	11	anbi2d 636	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))) ↔ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
13	12	rexbidv 3163	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
14	13	2ralbidv 3203	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
15		islly 23451	. . . . . . . 8 ⊢ (𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
16	15	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
17	4, 14, 16	3bitr4rd 313	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)))))
18	17	pm5.32da 584	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))
19	18	anbi2d 636	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)))))))
20		3anass 1100	. . . 4 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
21		3anass 1100	. . . 4 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)))) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))
22	19, 20, 21	3bitr4g 315	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))
23	22	adantr 481	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))
24	1, 23	bitrd 280	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∩ cin 3882  𝒫 cpw 4529   class class class wbr 5072  Rel wrel 5623  ‘cfv 6485  (class class class)co 7356   Er wer 8630  [cec 8631  ℕ₀cn0 12428   ↾_t crest 17374  TopOpenctopn 17375  Topctop 22876  Hauscha 23291  2^ndωc2ndc 23421  Locally clly 23447   ≃ chmph 23737  𝔼_hilcehl 25369  ManTopcmntop 34206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-1o 8395  df-er 8633  df-ec 8635  df-map 8765  df-top 22877  df-topon 22894  df-cn 23210  df-haus 23298  df-lly 23449  df-hmeo 23738  df-hmph 23739  df-mntop 34207

This theorem is referenced by: (None)