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Theorem nlly2i 20267
Description: Eliminate the neighborhood symbol from nllyi 20266. (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  /\  ( Jt  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 20266 . 2  |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  E. s  e.  ( ( nei `  J
) `  { P } ) ( s 
C_  U  /\  ( Jt  s )  e.  A
) )
2 simprrl 766 . . . . . 6  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  ->  s  C_  U
)
3 selpw 3961 . . . . . 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  /\  ( Jt  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  /\  ( Jt  s )  e.  A ) ) )  ->  J  e. 𝑛Locally  A )
6 nllytop 20264 . . . . . . . 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  /\  ( Jt  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  /\  ( Jt  s )  e.  A ) ) )  ->  s  e.  ( ( nei `  J
) `  { P } ) )
9 neii2 19900 . . . . . . 7  |-  ( ( J  e.  Top  /\  s  e.  ( ( nei `  J ) `  { P } ) )  ->  E. u  e.  J  ( { P }  C_  u  /\  u  C_  s
) )
107, 8, 9syl2anc 659 . . . . . 6  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  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  /\  ( Jt  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  /\  ( Jt  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  /\  ( Jt  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  /\  ( Jt  s )  e.  A ) ) )  /\  ( { P }  C_  u  /\  u  C_  s ) )  ->  P  e.  u )
16 simprr 758 . . . . . . . . 9  |-  ( ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U
)  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  /\  ( { P }  C_  u  /\  u  C_  s ) )  ->  u  C_  s )
17 simprrr 767 . . . . . . . . . 10  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  ->  ( Jt  s )  e.  A )
1817adantr 463 . . . . . . . . 9  |-  ( ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U
)  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  /\  ( { P }  C_  u  /\  u  C_  s ) )  -> 
( Jt  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  /\  ( Jt  s )  e.  A ) ) )  /\  ( { P }  C_  u  /\  u  C_  s ) )  -> 
( P  e.  u  /\  u  C_  s  /\  ( Jt  s )  e.  A ) )
2019ex 432 . . . . . . 7  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  ->  ( ( { P }  C_  u  /\  u  C_  s )  ->  ( P  e.  u  /\  u  C_  s  /\  ( Jt  s )  e.  A ) ) )
2120reximdv 2877 . . . . . 6  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  ->  ( E. u  e.  J  ( { P }  C_  u  /\  u  C_  s )  ->  E. u  e.  J  ( P  e.  u  /\  u  C_  s  /\  ( Jt  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  /\  ( Jt  s )  e.  A ) ) )  ->  E. u  e.  J  ( P  e.  u  /\  u  C_  s  /\  ( Jt  s )  e.  A ) )
234, 22jca 530 . . . 4  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  /\  ( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) ) )  ->  ( s  e. 
~P U  /\  E. u  e.  J  ( P  e.  u  /\  u  C_  s  /\  ( Jt  s )  e.  A
) ) )
2423ex 432 . . 3  |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  (
( s  e.  ( ( nei `  J
) `  { P } )  /\  (
s  C_  U  /\  ( Jt  s )  e.  A ) )  -> 
( s  e.  ~P U  /\  E. u  e.  J  ( P  e.  u  /\  u  C_  s  /\  ( Jt  s )  e.  A ) ) ) )
2524reximdv2 2874 . 2  |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  ( E. s  e.  (
( nei `  J
) `  { P } ) ( s 
C_  U  /\  ( Jt  s )  e.  A
)  ->  E. s  e.  ~P  U E. u  e.  J  ( P  e.  u  /\  u  C_  s  /\  ( Jt  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  /\  ( Jt  s )  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842   E.wrex 2754    C_ wss 3413   ~Pcpw 3954   {csn 3971   ` cfv 5568  (class class class)co 6277   ↾t crest 15033   Topctop 19684   neicnei 19889  𝑛Locally cnlly 20256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-top 19689  df-nei 19890  df-nlly 20258
This theorem is referenced by:  restnlly  20273  nllyrest  20277  nllyidm  20280  cldllycmp  20286  txnlly  20428  txkgen  20443  xkococnlem  20450  conpcon  29519  cvmliftmolem2  29566  cvmlift3lem8  29610
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