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Theorem nlly2i 19955
Description: Eliminate the neighborhood symbol from nllyi 19954. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nlly2i |- ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) -> E. s e. ~P U E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) )
Distinct variable groups:   u, s, A   P, s, u   U, s, u   J, s, u
Proof of Theorem nlly2i
StepHypRef Expression
1 nllyi 19954 . 2 |- ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) -> E. s e. ( ( nei ` J ) ` { P } ) ( s C_ U /\ ( J ↾t s ) e. A ) )
2 simprrl 765 . . . . . 6 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> s C_ U )
3 selpw 4004 . . . . . 6 |- ( s e. ~P U <-> s C_ U )
42, 3sylibr 212 . . . . 5 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> s e. ~P U )
5 simpl1 1000 . . . . . . . 8 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> J e. 𝑛Locally A )
6 nllytop 19952 . . . . . . . 8 |- ( J e. 𝑛Locally A -> J e. Top )
75, 6syl 16 . . . . . . 7 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> J e. Top )
8 simprl 756 . . . . . . 7 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> s e. ( ( nei ` J ) ` { P } ) )
9 neii2 19587 . . . . . . 7 |- ( ( J e. Top /\ s e. ( ( nei ` J ) ` { P } ) ) -> E. u e. J ( { P } C_ u /\ u C_ s ) )
107, 8, 9syl2anc 661 . . . . . 6 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> E. u e. J ( { P } C_ u /\ u C_ s ) )
11 simprl 756 . . . . . . . . . 10 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> { P } C_ u )
12 simpll3 1038 . . . . . . . . . . 11 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> P e. U )
13 snssg 4148 . . . . . . . . . . 11 |- ( P e. U -> ( P e. u <-> { P } C_ u ) )
1412, 13syl 16 . . . . . . . . . 10 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> ( P e. u <-> { P } C_ u ) )
1511, 14mpbird 232 . . . . . . . . 9 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> P e. u )
16 simprr 757 . . . . . . . . 9 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> u C_ s )
17 simprrr 766 . . . . . . . . . 10 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> ( J ↾t s ) e. A )
1817adantr 465 . . . . . . . . 9 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> ( J ↾t s ) e. A )
1915, 16, 183jca 1177 . . . . . . . 8 |- ( ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) /\ ( { P } C_ u /\ u C_ s ) ) -> ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) )
2019ex 434 . . . . . . 7 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> ( ( { P } C_ u /\ u C_ s ) -> ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) ) )
2120reximdv 2917 . . . . . 6 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> ( E. u e. J ( { P } C_ u /\ u C_ s ) -> E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) ) )
2210, 21mpd 15 . . . . 5 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) )
234, 22jca 532 . . . 4 |- ( ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) /\ ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) ) -> ( s e. ~P U /\ E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) ) )
2423ex 434 . . 3 |- ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) -> ( ( s e. ( ( nei ` J ) ` { P } ) /\ ( s C_ U /\ ( J ↾t s ) e. A ) ) -> ( s e. ~P U /\ E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) ) ) )
2524reximdv2 2914 . 2 |- ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) -> ( E. s e. ( ( nei ` J ) ` { P } ) ( s C_ U /\ ( J ↾t s ) e. A ) -> E. s e. ~P U E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) ) )
261, 25mpd 15 1 |- ( ( J e. 𝑛Locally A /\ U e. J /\ P e. U ) -> E. s e. ~P U E. u e. J ( P e. u /\ u C_ s /\ ( J ↾t s ) e. A ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   e. wcel 1804   E.wrex 2794   C_ wss 3461   ~Pcpw 3997   {csn 4014   ` cfv 5578  (class class class)co 6281   ↾t crest 14800   Topctop 19372   neicnei 19576  𝑛Locally cnlly 19944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-top 19377  df-nei 19577  df-nlly 19946
This theorem is referenced by:  restnlly  19961  nllyrest  19965  nllyidm  19968  cldllycmp  19974  txnlly  20116  txkgen  20131  xkococnlem  20138  conpcon  28658  cvmliftmolem2  28705  cvmlift3lem8  28749
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