Theorem ismntop 33989

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 33988	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
2		haustop 23355	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 532	. . . . . . 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 23808	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8753	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8791	. . . . . . . . . . . . 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 23810	. . . . . . . . . . . 12 ⊢ ((TopOpen‘(𝔼hil‘𝑁)) ≃ (𝐽 ↾t 𝑢) ↔ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁)))
10	8, 9	bitr2i 276	. . . . . . . . . . 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 630	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))) ↔ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
13	12	rexbidv 3177	. . . . . . . 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 3219	. . . . . . 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 23492	. . . . . . . 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 312	. . . . . 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 579	. . . . 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 630	. . . 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 1094	. . . 4 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
21		3anass 1094	. . . 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 314	. . 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 480	. 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 279	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 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∩ cin 3962  𝒫 cpw 4605   class class class wbr 5148  Rel wrel 5694  ‘cfv 6563  (class class class)co 7431   Er wer 8741  [cec 8742  ℕ₀cn0 12524   ↾_t crest 17467  TopOpenctopn 17468  Topctop 22915  Hauscha 23332  2^ndωc2ndc 23462  Locally clly 23488   ≃ chmph 23778  𝔼_hilcehl 25432  ManTopcmntop 33985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-ec 8746  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-haus 23339  df-lly 23490  df-hmeo 23779  df-hmph 23780  df-mntop 33986

This theorem is referenced by: (None)