Theorem ismntop 34183

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 34182	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
2		haustop 23275	. . . . . . . . 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 23728	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8644	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8682	. . . . . . . . . . . . 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 23730	. . . . . . . . . . . 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 3160	. . . . . . . 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 3200	. . . . . . 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 23412	. . . . . . . 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 2113  ∀wral 3051  ∃wrex 3060   ∩ cin 3900  𝒫 cpw 4554   class class class wbr 5098  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358   Er wer 8632  [cec 8633  ℕ₀cn0 12401   ↾_t crest 17340  TopOpenctopn 17341  Topctop 22837  Hauscha 23252  2^ndωc2ndc 23382  Locally clly 23408   ≃ chmph 23698  𝔼_hilcehl 25340  ManTopcmntop 34179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-1o 8397  df-er 8635  df-ec 8637  df-map 8765  df-top 22838  df-topon 22855  df-cn 23171  df-haus 23259  df-lly 23410  df-hmeo 23699  df-hmph 23700  df-mntop 34180

This theorem is referenced by: (None)