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Theorem nlly2i 19213
Description: Eliminate the neighborhood symbol from nllyi 19212. (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 19212 . 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 763 . . . . . 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 3976 . . . . . 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 991 . . . . . . . 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 19210 . . . . . . . 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 755 . . . . . . 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 18845 . . . . . . 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 755 . . . . . . . . . 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 1029 . . . . . . . . . . 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 4116 . . . . . . . . . . 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 756 . . . . . . . . 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 764 . . . . . . . . . 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 1168 . . . . . . . 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 2933 . . . . . 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 2931 . 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 965   e. wcel 1758   E.wrex 2800   C_ wss 3437   ~Pcpw 3969   {csn 3986   ` cfv 5527  (class class class)co 6201   ↾t crest 14479   Topctop 18631   neicnei 18834  𝑛Locally cnlly 19202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-top 18636  df-nei 18835  df-nlly 19204
This theorem is referenced by:  restnlly  19219  nllyrest  19223  nllyidm  19226  cldllycmp  19232  txnlly  19343  txkgen  19358  xkococnlem  19365  conpcon  27269  cvmliftmolem2  27316  cvmlift3lem8  27360
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