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Theorem llynlly 19744
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 19739 . 2  |-  ( J  e. Locally  A  ->  J  e. 
Top )
2 llyi 19741 . . . . 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 999 . . . . . . . . . . 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 16 . . . . . . . . . 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 755 . . . . . . . . . 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 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
) ) )  -> 
y  e.  u )
7 opnneip 19386 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  e.  J  /\  y  e.  u )  ->  u  e.  ( ( nei `  J ) `
 { y } ) )
84, 5, 6, 7syl3anc 1228 . . . . . . . . 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 1044 . . . . . . . . . 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 4017 . . . . . . . . . 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 3688 . . . . . . . 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 1046 . . . . . . . 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 532 . . . . . . 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 434 . . . . . 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 2934 . . . . 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 1197 . . 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 2885 . 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 19736 . 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 664 1  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   ` cfv 5586  (class class class)co 6282   ↾t crest 14672   Topctop 19161   neicnei 19364  Locally clly 19731  𝑛Locally cnlly 19732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-top 19166  df-nei 19365  df-lly 19733  df-nlly 19734
This theorem is referenced by:  llyssnlly  19745  symgtgp  20335
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