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Theorem llynlly 17454
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 17449 . 2  |-  ( J  e. Locally  A  ->  J  e. 
Top )
2 llyi 17451 . . . . 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 960 . . . . . . . . . . 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 733 . . . . . . . . . 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 1006 . . . . . . . . . 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 17099 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  e.  J  /\  y  e.  u )  ->  u  e.  ( ( nei `  J ) `
 { y } ) )
84, 5, 6, 7syl3anc 1184 . . . . . . . . 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 1005 . . . . . . . . . 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 2895 . . . . . . . . . . 11  |-  u  e. 
_V
1110elpw 3741 . . . . . . . . . 10  |-  ( u  e.  ~P x  <->  u  C_  x
)
129, 11sylibr 204 . . . . . . . . 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 3466 . . . . . . . . 9  |-  ( u  e.  ( ( ( nei `  J ) `
 { y } )  i^i  ~P x
)  <->  ( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P x ) )
148, 12, 13sylanbrc 646 . . . . . . . 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 1007 . . . . . . . 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 519 . . . . . . 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 424 . . . . . 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 2751 . . . . 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 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 )
20193expb 1154 . . 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 2734 . 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 17446 . 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 646 1  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1717   A.wral 2642   E.wrex 2643    i^i cin 3255    C_ wss 3256   ~Pcpw 3735   {csn 3750   ` cfv 5387  (class class class)co 6013   ↾t crest 13568   Topctop 16874   neicnei 17077  Locally clly 17441  𝑛Locally cnlly 17442
This theorem is referenced by:  llyssnlly  17455  symgtgp  18045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-top 16879  df-nei 17078  df-lly 17443  df-nlly 17444
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