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Theorem llyidm 19092
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 19076 . . . 4  |-  ( j  e. Locally Locally  A  ->  j  e.  Top )
2 llyi 19078 . . . . . . 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 1038 . . . . . . . . 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 755 . . . . . . . . . 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 3375 . . . . . . . . . . 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 1009 . . . . . . . . . . . 12  |-  ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  ->  j  e.  Top )
87adantr 465 . . . . . . . . . . 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 18781 . . . . . . . . . . 11  |-  ( ( j  e.  Top  /\  u  e.  j )  ->  ( u  e.  ( jt  u )  <->  ( u  e.  j  /\  u  C_  u ) ) )
108, 4, 9syl2anc 661 . . . . . . . . . 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 913 . . . . . . . . 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 1037 . . . . . . . . 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 19078 . . . . . . . . 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 1218 . . . . . . . 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 18781 . . . . . . . . . . . 12  |-  ( ( j  e.  Top  /\  u  e.  j )  ->  ( v  e.  ( jt  u )  <->  ( v  e.  j  /\  v  C_  u ) ) )
168, 4, 15syl2anc 661 . . . . . . . . . . 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 457 . . . . . . . . . . 11  |-  ( ( v  e.  j  /\  v  C_  u )  -> 
v  e.  j )
1816, 17syl6bi 228 . . . . . . . . . 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 755 . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 3366 . . . . . . . . . . . . . 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 selpw 3867 . . . . . . . . . . . . . 14  |-  ( v  e.  ~P x  <->  v  C_  x )
2523, 24sylibr 212 . . . . . . . . . . . . 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 )
2619, 25elind 3540 . . . . . . . . . . . 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 ) )
27 simprr2 1037 . . . . . . . . . . . 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 )
288adantr 465 . . . . . . . . . . . . . 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 )
29 simplrl 759 . . . . . . . . . . . . . 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 )
30 restabs 18769 . . . . . . . . . . . . . 14  |-  ( ( j  e.  Top  /\  v  C_  u  /\  u  e.  j )  ->  (
( jt  u )t  v )  =  ( jt  v ) )
3128, 20, 29, 30syl3anc 1218 . . . . . . . . . . . . 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 ) )
32 simprr3 1038 . . . . . . . . . . . . 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 )
3331, 32eqeltrrd 2518 . . . . . . . . . . . 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
)
3426, 27, 33jca32 535 . . . . . . . . . . 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 ) ) )
3534ex 434 . . . . . . . . . 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 ) ) ) )
3618, 35syland 481 . . . . . . . . 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 ) ) ) )
3736reximdv2 2825 . . . . . . . 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
) ) )
3814, 37mpd 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
) )
392, 38rexlimddv 2845 . . . . . 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
) )
40393expb 1188 . . . . 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
) )
4140ralrimivva 2808 . . . 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
) )
42 islly 19072 . . . 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
) ) )
431, 41, 42sylanbrc 664 . . 3  |-  ( j  e. Locally Locally  A  ->  j  e. Locally  A )
4443ssriv 3360 . 2  |- Locally Locally  A  C_ Locally  A
45 llyrest 19089 . . . . 5  |-  ( ( j  e. Locally  A  /\  x  e.  j )  ->  ( jt  x )  e. Locally  A )
4645adantl 466 . . . 4  |-  ( ( T.  /\  ( j  e. Locally  A  /\  x  e.  j ) )  -> 
( jt  x )  e. Locally  A )
47 llytop 19076 . . . . . 6  |-  ( j  e. Locally  A  ->  j  e. 
Top )
4847ssriv 3360 . . . . 5  |- Locally  A  C_  Top
4948a1i 11 . . . 4  |-  ( T. 
-> Locally  A  C_  Top )
5046, 49restlly 19087 . . 3  |-  ( T. 
-> Locally  A  C_ Locally Locally  A )
5150trud 1378 . 2  |- Locally  A  C_ Locally Locally  A
5244, 51eqssi 3372 1  |- Locally Locally  A  = Locally  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   T. wtru 1370    e. wcel 1756   A.wral 2715   E.wrex 2716    i^i cin 3327    C_ wss 3328   ~Pcpw 3860  (class class class)co 6091   ↾t crest 14359   Topctop 18498  Locally clly 19068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-oadd 6924  df-er 7101  df-en 7311  df-fin 7314  df-fi 7661  df-rest 14361  df-topgen 14382  df-top 18503  df-bases 18505  df-topon 18506  df-lly 19070
This theorem is referenced by: (None)
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