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Theorem nlly2i 17544
Description: Eliminate the neighborhood symbol from nllyi 17543. (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 17543 . 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 742 . . . . . 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 vex 2961 . . . . . . 7  |-  s  e. 
_V
43elpw 3807 . . . . . 6  |-  ( s  e.  ~P U  <->  s  C_  U )
52, 4sylibr 205 . . . . 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 )
6 simpl1 961 . . . . . . . 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 )
7 nllytop 17541 . . . . . . . 8  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
86, 7syl 16 . . . . . . 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 )
9 simprl 734 . . . . . . 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 } ) )
10 neii2 17177 . . . . . . 7  |-  ( ( J  e.  Top  /\  s  e.  ( ( nei `  J ) `  { P } ) )  ->  E. u  e.  J  ( { P }  C_  u  /\  u  C_  s
) )
118, 9, 10syl2anc 644 . . . . . 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
) )
12 simprl 734 . . . . . . . . . 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
)
13 simpll3 999 . . . . . . . . . . 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 )
14 snssg 3934 . . . . . . . . . . 11  |-  ( P  e.  U  ->  ( P  e.  u  <->  { P }  C_  u ) )
1513, 14syl 16 . . . . . . . . . 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
) )
1612, 15mpbird 225 . . . . . . . . 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 )
17 simprr 735 . . . . . . . . 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 )
18 simprrr 743 . . . . . . . . . 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 )
1918adantr 453 . . . . . . . . 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 )
2016, 17, 193jca 1135 . . . . . . . 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 ) )
2120ex 425 . . . . . . 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 ) ) )
2221reximdv 2819 . . . . . 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 ) ) )
2311, 22mpd 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 ) )
245, 23jca 520 . . . 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
) ) )
2524ex 425 . . 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 ) ) ) )
2625reximdv2 2817 . 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 ) ) )
271, 26mpd 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   E.wrex 2708    C_ wss 3322   ~Pcpw 3801   {csn 3816   ` cfv 5457  (class class class)co 6084   ↾t crest 13653   Topctop 16963   neicnei 17166  𝑛Locally cnlly 17533
This theorem is referenced by:  restnlly  17550  nllyrest  17554  nllyidm  17557  cldllycmp  17563  txnlly  17674  txkgen  17689  xkococnlem  17696  conpcon  24927  cvmliftmolem2  24974  cvmlift3lem8  25018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-top 16968  df-nei 17167  df-nlly 17535
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