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Theorem llynlly 20270
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 20265 . 2  |-  ( J  e. Locally  A  ->  J  e. 
Top )
2 llyi 20267 . . . . 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 1000 . . . . . . . . . . 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 17 . . . . . . . . . 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 756 . . . . . . . . . 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 1046 . . . . . . . . . 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 19913 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  e.  J  /\  y  e.  u )  ->  u  e.  ( ( nei `  J ) `
 { y } ) )
84, 5, 6, 7syl3anc 1230 . . . . . . . . 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 1045 . . . . . . . . . 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 selpw 3962 . . . . . . . . . 10  |-  ( u  e.  ~P x  <->  u  C_  x
)
119, 10sylibr 212 . . . . . . . . 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
)
128, 11elind 3627 . . . . . . . 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 ) )
13 simprr3 1047 . . . . . . . 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
)
1412, 13jca 530 . . . . . . 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
) )
1514ex 432 . . . . . 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
) ) )
1615reximdv2 2875 . . . . 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
) )
172, 16mpd 15 . . . 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 )
18173expb 1198 . . 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
)
1918ralrimivva 2825 . 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
)
20 isnlly 20262 . 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
) )
211, 19, 20sylanbrc 662 1  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    e. wcel 1842   A.wral 2754   E.wrex 2755    i^i cin 3413    C_ wss 3414   ~Pcpw 3955   {csn 3972   ` cfv 5569  (class class class)co 6278   ↾t crest 15035   Topctop 19686   neicnei 19891  Locally clly 20257  𝑛Locally cnlly 20258
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
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 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-top 19691  df-nei 19892  df-lly 20259  df-nlly 20260
This theorem is referenced by:  llyssnlly  20271  symgtgp  20892
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