Theorem ismntop 34204

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 34203	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
2		haustop 23287	. . . . . . . . 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 23740	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8655	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8693	. . . . . . . . . . . . 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 23742	. . . . . . . . . . . 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 631	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ Haus) → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))) ↔ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
13	12	rexbidv 3162	. . . . . . . 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 3202	. . . . . . 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 23424	. . . . . . . 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 631	. . . 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 1095	. . . 4 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
21		3anass 1095	. . . 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 1087   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902  𝒫 cpw 4556   class class class wbr 5100  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368   Er wer 8642  [cec 8643  ℕ₀cn0 12413   ↾_t crest 17352  TopOpenctopn 17353  Topctop 22849  Hauscha 23264  2^ndωc2ndc 23394  Locally clly 23420   ≃ chmph 23710  𝔼_hilcehl 25352  ManTopcmntop 34200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-1o 8407  df-er 8645  df-ec 8647  df-map 8777  df-top 22850  df-topon 22867  df-cn 23183  df-haus 23271  df-lly 23422  df-hmeo 23711  df-hmph 23712  df-mntop 34201

This theorem is referenced by: (None)