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Theorem nlly2i 17308
Description: Eliminate the neighborhood symbol from nllyi 17307. (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 17307 . 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 2867 . . . . . . 7  |-  s  e. 
_V
43elpw 3707 . . . . . 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 17305 . . . . . . . 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 16951 . . . . . . 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 3830 . . . . . . . . . . 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 2730 . . . . . 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 2728 . 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 1710   E.wrex 2620    C_ wss 3228   ~Pcpw 3701   {csn 3716   ` cfv 5337  (class class class)co 5945   ↾t crest 13424   Topctop 16737   neicnei 16940  𝑛Locally cnlly 17297
This theorem is referenced by:  restnlly  17314  nllyrest  17318  nllyidm  17321  cldllycmp  17327  txnlly  17437  txkgen  17452  xkococnlem  17459  conpcon  24170  cvmliftmolem2  24217  cvmlift3lem8  24261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-top 16742  df-nei 16941  df-nlly 17299
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