Theorem nlly2i 23459

Description: Eliminate the neighborhood symbol from nllyi 23458. (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 23458	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ ((nei‘𝐽)‘{𝑃})(𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
2		simprrl 786	. . 3 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑈)
3		velpw 4534	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑈 ↔ 𝑠 ⊆ 𝑈)
4	2, 3	sylibr 235	. 2 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑈)
5		simpl1 1198	. . . . 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 23456	. . . . 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 776	. . . 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 23091	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ ((nei‘𝐽)‘{𝑃})) → ∃𝑢 ∈ 𝐽 ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠))
10	7, 8, 9	syl2anc 590	. . 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 776	. . . . . . 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 1221	. . . . . . . 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 4715	. . . . . . . 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 258	. . . . . 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 778	. . . . . 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 787	. . . . . . 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 481	. . . . . 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 1134	. . . . 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 413	. . . 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 3154	. . 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 3157	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 207   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ‘cfv 6485  (class class class)co 7356   ↾_t crest 17374  Topctop 22876  neicnei 23080  𝑛-Locally cnlly 23448

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-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

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-reu 3345  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-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-top 22877  df-nei 23081  df-nlly 23450

This theorem is referenced by:  restnlly  23465  nllyrest  23469  nllyidm  23472  cldllycmp  23478  txnlly  23620  txkgen  23635  xkococnlem  23642  connpconn  35463  cvmliftmolem2  35510  cvmlift3lem8  35554