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Theorem llynlly 17203
Description: A locally  A space is n-locally  A: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )

Proof of Theorem llynlly
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 17198 . 2  |-  ( J  e. Locally  A  ->  J  e. 
Top )
2 llyi 17200 . . . . 5  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  E. u  e.  J  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) )
3 simpl1 958 . . . . . . . . . . 11  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  J  e. Locally  A )
43, 1syl 15 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  J  e.  Top )
5 simprl 732 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  J )
6 simprr2 1004 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  -> 
y  e.  u )
7 opnneip 16856 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  e.  J  /\  y  e.  u )  ->  u  e.  ( ( nei `  J ) `
 { y } ) )
84, 5, 6, 7syl3anc 1182 . . . . . . . . 9  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  ( ( nei `  J ) `  { y } ) )
9 simprr1 1003 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  C_  x )
10 vex 2791 . . . . . . . . . . 11  |-  u  e. 
_V
1110elpw 3631 . . . . . . . . . 10  |-  ( u  e.  ~P x  <->  u  C_  x
)
129, 11sylibr 203 . . . . . . . . 9  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  ~P x
)
13 elin 3358 . . . . . . . . 9  |-  ( u  e.  ( ( ( nei `  J ) `
 { y } )  i^i  ~P x
)  <->  ( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P x ) )
148, 12, 13sylanbrc 645 . . . . . . . 8  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  ( (
( nei `  J
) `  { y } )  i^i  ~P x ) )
15 simprr3 1005 . . . . . . . 8  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  -> 
( Jt  u )  e.  A
)
1614, 15jca 518 . . . . . . 7  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  -> 
( u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P x )  /\  ( Jt  u )  e.  A
) )
1716ex 423 . . . . . 6  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  ( ( u  e.  J  /\  ( u 
C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) )  ->  (
u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P x )  /\  ( Jt  u )  e.  A
) ) )
1817reximdv2 2652 . . . . 5  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  ( E. u  e.  J  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
)  ->  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
192, 18mpd 14 . . . 4  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  E. u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) ( Jt  u )  e.  A )
20193expb 1152 . . 3  |-  ( ( J  e. Locally  A  /\  ( x  e.  J  /\  y  e.  x
) )  ->  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
)
2120ralrimivva 2635 . 2  |-  ( J  e. Locally  A  ->  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
)
22 isnlly 17195 . 2  |-  ( J  e. 𝑛Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
231, 21, 22sylanbrc 645 1  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   neicnei 16834  Locally clly 17190  𝑛Locally cnlly 17191
This theorem is referenced by:  llyssnlly  17204  symgtgp  17784
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-lly 17192  df-nlly 17193
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