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Theorem llyidm 20580
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 20564 . . . 4  |-  ( j  e. Locally Locally  A  ->  j  e.  Top )
2 llyi 20566 . . . . . . 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 1080 . . . . . . . . 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 772 . . . . . . . . . 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 3437 . . . . . . . . . . 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 1051 . . . . . . . . . . . 12  |-  ( ( j  e. Locally Locally  A  /\  x  e.  j  /\  y  e.  x )  ->  j  e.  Top )
87adantr 472 . . . . . . . . . . 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 20270 . . . . . . . . . . 11  |-  ( ( j  e.  Top  /\  u  e.  j )  ->  ( u  e.  ( jt  u )  <->  ( u  e.  j  /\  u  C_  u ) ) )
108, 4, 9syl2anc 673 . . . . . . . . . 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 936 . . . . . . . . 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 1079 . . . . . . . . 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 20566 . . . . . . . . 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 1292 . . . . . . . 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 20270 . . . . . . . . . . . 12  |-  ( ( j  e.  Top  /\  u  e.  j )  ->  ( v  e.  ( jt  u )  <->  ( v  e.  j  /\  v  C_  u ) ) )
168, 4, 15syl2anc 673 . . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( ( v  e.  j  /\  v  C_  u )  -> 
v  e.  j )
1816, 17syl6bi 236 . . . . . . . . . 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 772 . . . . . . . . . . . . 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 1078 . . . . . . . . . . . . . . 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 1078 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 3428 . . . . . . . . . . . . . 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 3949 . . . . . . . . . . . . . 14  |-  ( v  e.  ~P x  <->  v  C_  x )
2523, 24sylibr 217 . . . . . . . . . . . . 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 3609 . . . . . . . . . . . 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 1079 . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 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 20258 . . . . . . . . . . . . . 14  |-  ( ( j  e.  Top  /\  v  C_  u  /\  u  e.  j )  ->  (
( jt  u )t  v )  =  ( jt  v ) )
3128, 20, 29, 30syl3anc 1292 . . . . . . . . . . . . 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 1080 . . . . . . . . . . . . 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 2550 . . . . . . . . . . . 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 544 . . . . . . . . . . 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 441 . . . . . . . . . 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 489 . . . . . . . . 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 2855 . . . . . . . 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 2875 . . . . . 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 1232 . . . . 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 2814 . . . 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 20560 . . . 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 677 . . 3  |-  ( j  e. Locally Locally  A  ->  j  e. Locally  A )
4443ssriv 3422 . 2  |- Locally Locally  A  C_ Locally  A
45 llyrest 20577 . . . . 5  |-  ( ( j  e. Locally  A  /\  x  e.  j )  ->  ( jt  x )  e. Locally  A )
4645adantl 473 . . . 4  |-  ( ( T.  /\  ( j  e. Locally  A  /\  x  e.  j ) )  -> 
( jt  x )  e. Locally  A )
47 llytop 20564 . . . . . 6  |-  ( j  e. Locally  A  ->  j  e. 
Top )
4847ssriv 3422 . . . . 5  |- Locally  A  C_  Top
4948a1i 11 . . . 4  |-  ( T. 
-> Locally  A  C_  Top )
5046, 49restlly 20575 . . 3  |-  ( T. 
-> Locally  A  C_ Locally Locally  A )
5150trud 1461 . 2  |- Locally  A  C_ Locally Locally  A
5244, 51eqssi 3434 1  |- Locally Locally  A  = Locally  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   T. wtru 1453    e. wcel 1904   A.wral 2756   E.wrex 2757    i^i cin 3389    C_ wss 3390   ~Pcpw 3942  (class class class)co 6308   ↾t crest 15397   Topctop 19994  Locally clly 20556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-lly 20558
This theorem is referenced by: (None)
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