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Theorem nlly2i 20484
Description: Eliminate the neighborhood symbol from nllyi 20483. (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 20483 . 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 773 . . . . . 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 3957 . . . . . 6 |- ( s e. ~P U <-> s C_ U )
42, 3sylibr 216 . . . . 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 1010 . . . . . . . 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 20481 . . . . . . . 8 |- ( J e. 𝑛Locally A -> J e. Top )
75, 6syl 17 . . . . . . 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 763 . . . . . . 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 20117 . . . . . . 7 |- ( ( J e. Top /\ s e. ( ( nei ` J ) ` { P } ) ) -> E. u e. J ( { P } C_ u /\ u C_ s ) )
107, 8, 9syl2anc 666 . . . . . 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 763 . . . . . . . . . 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 1048 . . . . . . . . . . 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 4104 . . . . . . . . . . 11 |- ( P e. U -> ( P e. u <-> { P } C_ u ) )
1412, 13syl 17 . . . . . . . . . 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 236 . . . . . . . . 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 765 . . . . . . . . 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 774 . . . . . . . . . 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 467 . . . . . . . . 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 1187 . . . . . . . 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 436 . . . . . . 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 2860 . . . . . 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 535 . . . 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 436 . . 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 2857 . 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 188   /\ wa 371   /\ w3a 984   e. wcel 1886   E.wrex 2737   C_ wss 3403   ~Pcpw 3950   {csn 3967   ` cfv 5581  (class class class)co 6288   ↾t crest 15312   Topctop 19910   neicnei 20106  𝑛Locally cnlly 20473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-top 19914  df-nei 20107  df-nlly 20475
This theorem is referenced by:  restnlly  20490  nllyrest  20494  nllyidm  20497  cldllycmp  20503  txnlly  20645  txkgen  20660  xkococnlem  20667  conpcon  29951  cvmliftmolem2  29998  cvmlift3lem8  30042
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