Theorem nlly2i 23432

Description: Eliminate the neighborhood symbol from nllyi 23431. (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 23431	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ ((nei‘𝐽)‘{𝑃})(𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
2		simprrl 781	. . 3 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑠 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑠 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑈)
3		velpw 4561	. . 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 1193	. . . . 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 23429	. . . . 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 771	. . . 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 23064	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ ((nei‘𝐽)‘{𝑃})) → ∃𝑢 ∈ 𝐽 ({𝑃} ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑠))
10	7, 8, 9	syl2anc 585	. . 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 771	. . . . . . 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 1216	. . . . . . . 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 4742	. . . . . . . 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 773	. . . . . 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 782	. . . . . . 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 1129	. . . . 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 3153	. . 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 3156	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 1087   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ‘cfv 6500  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  neicnei 23053  𝑛-Locally cnlly 23421

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

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-reu 3353  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-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-top 22850  df-nei 23054  df-nlly 23423

This theorem is referenced by:  restnlly  23438  nllyrest  23442  nllyidm  23445  cldllycmp  23451  txnlly  23593  txkgen  23608  xkococnlem  23615  connpconn  35451  cvmliftmolem2  35498  cvmlift3lem8  35542