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Theorem llyidm 20580
 Description: Idempotence of the "locally" predicate, i.e. being "locally " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyidm Locally Locally Locally

Proof of Theorem llyidm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 20564 . . . 4 Locally Locally
2 llyi 20566 . . . . . . 7 Locally Locally t Locally
3 simprr3 1080 . . . . . . . . 9 Locally Locally t Locally t Locally
4 simprl 772 . . . . . . . . . 10 Locally Locally t Locally
5 ssid 3437 . . . . . . . . . . 11
65a1i 11 . . . . . . . . . 10 Locally Locally t Locally
713ad2ant1 1051 . . . . . . . . . . . 12 Locally Locally
87adantr 472 . . . . . . . . . . 11 Locally Locally t Locally
9 restopn2 20270 . . . . . . . . . . 11 t
108, 4, 9syl2anc 673 . . . . . . . . . 10 Locally Locally t Locally t
114, 6, 10mpbir2and 936 . . . . . . . . 9 Locally Locally t Locally t
12 simprr2 1079 . . . . . . . . 9 Locally Locally t Locally
13 llyi 20566 . . . . . . . . 9 t Locally t t t t
143, 11, 12, 13syl3anc 1292 . . . . . . . 8 Locally Locally t Locally t t t
15 restopn2 20270 . . . . . . . . . . . 12 t
168, 4, 15syl2anc 673 . . . . . . . . . . 11 Locally Locally t Locally t
17 simpl 464 . . . . . . . . . . 11
1816, 17syl6bi 236 . . . . . . . . . 10 Locally Locally t Locally t
19 simprl 772 . . . . . . . . . . . . 13 Locally Locally t Locally t t
20 simprr1 1078 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t
21 simprr1 1078 . . . . . . . . . . . . . . . 16 Locally Locally t Locally
2221adantr 472 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t
2320, 22sstrd 3428 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
24 selpw 3949 . . . . . . . . . . . . . 14
2523, 24sylibr 217 . . . . . . . . . . . . 13 Locally Locally t Locally t t
2619, 25elind 3609 . . . . . . . . . . . 12 Locally Locally t Locally t t
27 simprr2 1079 . . . . . . . . . . . 12 Locally Locally t Locally t t
288adantr 472 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
29 simplrl 778 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
30 restabs 20258 . . . . . . . . . . . . . 14 t t t
3128, 20, 29, 30syl3anc 1292 . . . . . . . . . . . . 13 Locally Locally t Locally t t t t t
32 simprr3 1080 . . . . . . . . . . . . 13 Locally Locally t Locally t t t t
3331, 32eqeltrrd 2550 . . . . . . . . . . . 12 Locally Locally t Locally t t t
3426, 27, 33jca32 544 . . . . . . . . . . 11 Locally Locally t Locally t t t
3534ex 441 . . . . . . . . . 10 Locally Locally t Locally t t t
3618, 35syland 489 . . . . . . . . 9 Locally Locally t Locally t t t t
3736reximdv2 2855 . . . . . . . 8 Locally Locally t Locally t t t t
3814, 37mpd 15 . . . . . . 7 Locally Locally t Locally t
392, 38rexlimddv 2875 . . . . . 6 Locally Locally t
40393expb 1232 . . . . 5 Locally Locally t
4140ralrimivva 2814 . . . 4 Locally Locally t
42 islly 20560 . . . 4 Locally t
431, 41, 42sylanbrc 677 . . 3 Locally Locally Locally
4443ssriv 3422 . 2 Locally Locally Locally
45 llyrest 20577 . . . . 5 Locally t Locally
4645adantl 473 . . . 4 Locally t Locally
47 llytop 20564 . . . . . 6 Locally
4847ssriv 3422 . . . . 5 Locally
4948a1i 11 . . . 4 Locally
5046, 49restlly 20575 . . 3 Locally Locally Locally
5150trud 1461 . 2 Locally Locally Locally
5244, 51eqssi 3434 1 Locally Locally Locally
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376   w3a 1007   wceq 1452   wtru 1453   wcel 1904  wral 2756  wrex 2757   cin 3389   wss 3390  cpw 3942  (class class class)co 6308   ↾t crest 15397  ctop 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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