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Theorem nlly2i 21279
Description: Eliminate the neighborhood symbol from nllyi 21278. (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 21278 . 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 804 . . . . . 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 4165 . . . . . 6 |- ( s e. ~P U <-> s C_ U )
42, 3sylibr 224 . . . . 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 1064 . . . . . . . 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 21276 . . . . . . . 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 794 . . . . . . 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 20912 . . . . . . 7 |- ( ( J e. Top /\ s e. ( ( nei ` J ) ` { P } ) ) -> E. u e. J ( { P } C_ u /\ u C_ s ) )
107, 8, 9syl2anc 693 . . . . . 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 794 . . . . . . . . . 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 1102 . . . . . . . . . . 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 4327 . . . . . . . . . . 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 247 . . . . . . . . 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 796 . . . . . . . . 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 805 . . . . . . . . . 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 481 . . . . . . . . 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 1242 . . . . . . . 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 450 . . . . . . 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 3016 . . . . . 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 554 . . . 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 450 . . 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 3014 . 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 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   {csn 4177   ` cfv 5888  (class class class)co 6650   ↾t crest 16081   Topctop 20698   neicnei 20901  𝑛Locally cnlly 21268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-top 20699  df-nei 20902  df-nlly 21270
This theorem is referenced by:  restnlly  21285  nllyrest  21289  nllyidm  21292  cldllycmp  21298  txnlly  21440  txkgen  21455  xkococnlem  21462  connpconn  31217  cvmliftmolem2  31264  cvmlift3lem8  31308
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