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Theorem nlly2i 17202
Description: Eliminate the neighborhood symbol from nllyi 17201. (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 17201 . 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 740 . . . . . 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 2791 . . . . . . 7  |-  s  e. 
_V
43elpw 3631 . . . . . 6  |-  ( s  e.  ~P U  <->  s  C_  U )
52, 4sylibr 203 . . . . 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 958 . . . . . . . 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 17199 . . . . . . . 8  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
86, 7syl 15 . . . . . . 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 732 . . . . . . 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 16845 . . . . . . 7  |-  ( ( J  e.  Top  /\  s  e.  ( ( nei `  J ) `  { P } ) )  ->  E. u  e.  J  ( { P }  C_  u  /\  u  C_  s
) )
118, 9, 10syl2anc 642 . . . . . 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 732 . . . . . . . . . 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 996 . . . . . . . . . . 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 3754 . . . . . . . . . . 11  |-  ( P  e.  U  ->  ( P  e.  u  <->  { P }  C_  u ) )
1513, 14syl 15 . . . . . . . . . 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 223 . . . . . . . . 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 733 . . . . . . . . 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 741 . . . . . . . . . 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 451 . . . . . . . . 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 1132 . . . . . . . 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 423 . . . . . . 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 2654 . . . . . 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 14 . . . . 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 518 . . . 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 423 . . 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 2652 . 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 14 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 176    /\ wa 358    /\ w3a 934    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   neicnei 16834  𝑛Locally cnlly 17191
This theorem is referenced by:  restnlly  17208  nllyrest  17212  nllyidm  17215  cldllycmp  17221  txnlly  17331  txkgen  17346  xkococnlem  17353  conpcon  23766  cvmliftmolem2  23813  cvmlift3lem8  23857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-top 16636  df-nei 16835  df-nlly 17193
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