Theorem nlly2i 23339

Description: Eliminate the neighborhood symbol from nllyi 23338. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nlly2i	⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ 𝒫 𝑈∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))

Distinct variable groups:   𝑢,𝑠,𝐴   𝑃,𝑠,𝑢   𝑈,𝑠,𝑢   𝐽,𝑠,𝑢

Proof of Theorem nlly2i

Step	Hyp	Ref	Expression
1		nllyi 23338	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ ((nei‘𝐽)‘{𝑃})(𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
2		simprrl 780	. . 3 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑈)
3		velpw 4564	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑈 ↔ 𝑠 ⊆ 𝑈)
4	2, 3	sylibr 234	. 2 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑈)
5		simpl1 1192	. . . . 5 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐽 ∈ 𝑛-Locally 𝐴)
6		nllytop 23336	. . . . 5 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)
7	5, 6	syl 17	. . . 4 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐽 ∈ Top)
8		simprl 770	. . . 4 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
9		neii2 22971	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ ((nei‘𝐽)‘{𝑃})) → ∃𝑢 ∈ 𝐽 ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠))
10	7, 8, 9	syl2anc 584	. . 3 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → ∃𝑢 ∈ 𝐽 ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠))
11		simprl 770	. . . . . . 7 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → {𝑃} ⊆ 𝑢)
12		simpll3 1215	. . . . . . . 8 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → 𝑃 ∈ 𝑈)
13		snssg 4743	. . . . . . . 8 ⊢ (𝑃 ∈ 𝑈 → (𝑃 ∈ 𝑢 ↔ {𝑃} ⊆ 𝑢))
14	12, 13	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → (𝑃 ∈ 𝑢 ↔ {𝑃} ⊆ 𝑢))
15	11, 14	mpbird 257	. . . . . 6 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → 𝑃 ∈ 𝑢)
16		simprr 772	. . . . . 6 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → 𝑢 ⊆ 𝑠)
17		simprrr 781	. . . . . . 7 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (𝐽 ↾t 𝑠) ∈ 𝐴)
18	17	adantr 480	. . . . . 6 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → (𝐽 ↾t 𝑠) ∈ 𝐴)
19	15, 16, 18	3jca 1128	. . . . 5 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) ∧ ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠)) → (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
20	19	ex 412	. . . 4 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠) → (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴)))
21	20	reximdv 3148	. . 3 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (∃𝑢 ∈ 𝐽 ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠) → ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴)))
22	10, 21	mpd 15	. 2 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
23	1, 4, 22	reximssdv 3151	1 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ 𝒫 𝑈∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ‘cfv 6499  (class class class)co 7369   ↾_t crest 17359  Topctop 22756  neicnei 22960  𝑛-Locally cnlly 23328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-top 22757  df-nei 22961  df-nlly 23330

This theorem is referenced by:  restnlly  23345  nllyrest  23349  nllyidm  23352  cldllycmp  23358  txnlly  23500  txkgen  23515  xkococnlem  23522  connpconn  35195  cvmliftmolem2  35242  cvmlift3lem8  35286