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Theorem llyidm 17551
Description: Idempotence of the "locally" predicate, i.e. being "locally  A " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyidm  |- Locally Locally  A  = Locally  A

Proof of Theorem llyidm
Dummy variables  j  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 17535 . . . 4  |-  ( j  e. Locally Locally  A  ->  j  e.  Top )
2 llyi 17537 . . . . . . 7  |-  ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  ->  E. u  e.  j  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A )
)
3 simprr3 1007 . . . . . . . . 9  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  (
jt  u )  e. Locally  A )
4 simprl 733 . . . . . . . . . 10  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  u  e.  j )
5 ssid 3367 . . . . . . . . . . 11  |-  u  C_  u
65a1i 11 . . . . . . . . . 10  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  u  C_  u )
713ad2ant1 978 . . . . . . . . . . . 12  |-  ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  ->  j  e.  Top )
87adantr 452 . . . . . . . . . . 11  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  j  e.  Top )
9 restopn2 17241 . . . . . . . . . . 11  |-  ( ( j  e.  Top  /\  u  e.  j )  ->  ( u  e.  ( jt  u )  <->  ( u  e.  j  /\  u  C_  u ) ) )
108, 4, 9syl2anc 643 . . . . . . . . . 10  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  (
u  e.  ( jt  u )  <->  ( u  e.  j  /\  u  C_  u ) ) )
114, 6, 10mpbir2and 889 . . . . . . . . 9  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  u  e.  ( jt  u ) )
12 simprr2 1006 . . . . . . . . 9  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  y  e.  u )
13 llyi 17537 . . . . . . . . 9  |-  ( ( ( jt  u )  e. Locally  A  /\  u  e.  ( jt  u
)  /\  y  e.  u )  ->  E. v  e.  ( jt  u ) ( v 
C_  u  /\  y  e.  v  /\  (
( jt  u )t  v )  e.  A ) )
143, 11, 12, 13syl3anc 1184 . . . . . . . 8  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  E. v  e.  ( jt  u ) ( v 
C_  u  /\  y  e.  v  /\  (
( jt  u )t  v )  e.  A ) )
15 restopn2 17241 . . . . . . . . . . . 12  |-  ( ( j  e.  Top  /\  u  e.  j )  ->  ( v  e.  ( jt  u )  <->  ( v  e.  j  /\  v  C_  u ) ) )
168, 4, 15syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  (
v  e.  ( jt  u )  <->  ( v  e.  j  /\  v  C_  u ) ) )
17 simpl 444 . . . . . . . . . . 11  |-  ( ( v  e.  j  /\  v  C_  u )  -> 
v  e.  j )
1816, 17syl6bi 220 . . . . . . . . . 10  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  (
v  e.  ( jt  u )  ->  v  e.  j ) )
19 simprl 733 . . . . . . . . . . . . 13  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  v  e.  j )
20 simprr1 1005 . . . . . . . . . . . . . . 15  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  v  C_  u )
21 simprr1 1005 . . . . . . . . . . . . . . . 16  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  u  C_  x )
2221adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  u  C_  x )
2320, 22sstrd 3358 . . . . . . . . . . . . . 14  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  v  C_  x )
24 vex 2959 . . . . . . . . . . . . . . 15  |-  v  e. 
_V
2524elpw 3805 . . . . . . . . . . . . . 14  |-  ( v  e.  ~P x  <->  v  C_  x )
2623, 25sylibr 204 . . . . . . . . . . . . 13  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  v  e.  ~P x )
27 elin 3530 . . . . . . . . . . . . 13  |-  ( v  e.  ( j  i^i 
~P x )  <->  ( v  e.  j  /\  v  e.  ~P x ) )
2819, 26, 27sylanbrc 646 . . . . . . . . . . . 12  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  v  e.  ( j  i^i  ~P x ) )
29 simprr2 1006 . . . . . . . . . . . 12  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  y  e.  v )
308adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  j  e.  Top )
31 simplrl 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  u  e.  j )
32 restabs 17229 . . . . . . . . . . . . . 14  |-  ( ( j  e.  Top  /\  v  C_  u  /\  u  e.  j )  ->  (
( jt  u )t  v )  =  ( jt  v ) )
3330, 20, 31, 32syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  (
( jt  u )t  v )  =  ( jt  v ) )
34 simprr3 1007 . . . . . . . . . . . . 13  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  (
( jt  u )t  v )  e.  A )
3533, 34eqeltrrd 2511 . . . . . . . . . . . 12  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  (
jt  v )  e.  A
)
3628, 29, 35jca32 522 . . . . . . . . . . 11  |-  ( ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x
)  /\  ( u  e.  j  /\  (
u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  /\  (
v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) ) )  ->  (
v  e.  ( j  i^i  ~P x )  /\  ( y  e.  v  /\  ( jt  v )  e.  A ) ) )
3736ex 424 . . . . . . . . . 10  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  (
( v  e.  j  /\  ( v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A
) )  ->  (
v  e.  ( j  i^i  ~P x )  /\  ( y  e.  v  /\  ( jt  v )  e.  A ) ) ) )
3818, 37syland 468 . . . . . . . . 9  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  (
( v  e.  ( jt  u )  /\  (
v  C_  u  /\  y  e.  v  /\  ( ( jt  u )t  v )  e.  A ) )  ->  ( v  e.  ( j  i^i  ~P x )  /\  (
y  e.  v  /\  ( jt  v )  e.  A ) ) ) )
3938reximdv2 2815 . . . . . . . 8  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  ( E. v  e.  (
jt  u ) ( v 
C_  u  /\  y  e.  v  /\  (
( jt  u )t  v )  e.  A )  ->  E. v  e.  ( j  i^i  ~P x ) ( y  e.  v  /\  (
jt  v )  e.  A
) ) )
4014, 39mpd 15 . . . . . . 7  |-  ( ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  /\  ( u  e.  j  /\  ( u  C_  x  /\  y  e.  u  /\  ( jt  u )  e. Locally  A ) ) )  ->  E. v  e.  ( j  i^i  ~P x ) ( y  e.  v  /\  (
jt  v )  e.  A
) )
412, 40rexlimddv 2834 . . . . . 6  |-  ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  ->  E. v  e.  ( j  i^i  ~P x ) ( y  e.  v  /\  (
jt  v )  e.  A
) )
42413expb 1154 . . . . 5  |-  ( ( j  e. Locally Locally  A  /\  (
x  e.  j  /\  y  e.  x )
)  ->  E. v  e.  ( j  i^i  ~P x ) ( y  e.  v  /\  (
jt  v )  e.  A
) )
4342ralrimivva 2798 . . . 4  |-  ( j  e. Locally Locally  A  ->  A. x  e.  j  A. y  e.  x  E. v  e.  ( j  i^i  ~P x ) ( y  e.  v  /\  (
jt  v )  e.  A
) )
44 islly 17531 . . . 4  |-  ( j  e. Locally  A  <->  ( j  e. 
Top  /\  A. x  e.  j  A. y  e.  x  E. v  e.  ( j  i^i  ~P x ) ( y  e.  v  /\  (
jt  v )  e.  A
) ) )
451, 43, 44sylanbrc 646 . . 3  |-  ( j  e. Locally Locally  A  ->  j  e. Locally  A )
4645ssriv 3352 . 2  |- Locally Locally  A  C_ Locally  A
47 llyrest 17548 . . . . 5  |-  ( ( j  e. Locally  A  /\  x  e.  j )  ->  ( jt  x )  e. Locally  A )
4847adantl 453 . . . 4  |-  ( (  T.  /\  ( j  e. Locally  A  /\  x  e.  j ) )  -> 
( jt  x )  e. Locally  A )
49 llytop 17535 . . . . . 6  |-  ( j  e. Locally  A  ->  j  e. 
Top )
5049ssriv 3352 . . . . 5  |- Locally  A  C_  Top
5150a1i 11 . . . 4  |-  (  T. 
-> Locally  A  C_  Top )
5248, 51restlly 17546 . . 3  |-  (  T. 
-> Locally  A  C_ Locally Locally  A )
5352trud 1332 . 2  |- Locally  A  C_ Locally Locally  A
5446, 53eqssi 3364 1  |- Locally Locally  A  = Locally  A
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    T. wtru 1325    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   ~Pcpw 3799  (class class class)co 6081   ↾t crest 13648   Topctop 16958  Locally clly 17527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-oadd 6728  df-er 6905  df-en 7110  df-fin 7113  df-fi 7416  df-rest 13650  df-topgen 13667  df-top 16963  df-bases 16965  df-topon 16966  df-lly 17529
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