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Theorem llyidm 17214
 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 17198 . . . 4 Locally Locally
2 llyi 17200 . . . . . . 7 Locally Locally t Locally
3 simprr3 1005 . . . . . . . . . . 11 Locally Locally t Locally t Locally
4 simprl 732 . . . . . . . . . . . 12 Locally Locally t Locally
5 ssid 3197 . . . . . . . . . . . . 13
65a1i 10 . . . . . . . . . . . 12 Locally Locally t Locally
713ad2ant1 976 . . . . . . . . . . . . . 14 Locally Locally
87adantr 451 . . . . . . . . . . . . 13 Locally Locally t Locally
9 restopn2 16908 . . . . . . . . . . . . 13 t
108, 4, 9syl2anc 642 . . . . . . . . . . . 12 Locally Locally t Locally t
114, 6, 10mpbir2and 888 . . . . . . . . . . 11 Locally Locally t Locally t
12 simprr2 1004 . . . . . . . . . . 11 Locally Locally t Locally
13 llyi 17200 . . . . . . . . . . 11 t Locally t t t t
143, 11, 12, 13syl3anc 1182 . . . . . . . . . 10 Locally Locally t Locally t t t
15 restopn2 16908 . . . . . . . . . . . . . 14 t
168, 4, 15syl2anc 642 . . . . . . . . . . . . 13 Locally Locally t Locally t
17 simpl 443 . . . . . . . . . . . . 13
1816, 17syl6bi 219 . . . . . . . . . . . 12 Locally Locally t Locally t
19 simprl 732 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t
20 simprr1 1003 . . . . . . . . . . . . . . . . 17 Locally Locally t Locally t t
21 simprr1 1003 . . . . . . . . . . . . . . . . . 18 Locally Locally t Locally
2221adantr 451 . . . . . . . . . . . . . . . . 17 Locally Locally t Locally t t
2320, 22sstrd 3189 . . . . . . . . . . . . . . . 16 Locally Locally t Locally t t
24 vex 2791 . . . . . . . . . . . . . . . . 17
2524elpw 3631 . . . . . . . . . . . . . . . 16
2623, 25sylibr 203 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t
27 elin 3358 . . . . . . . . . . . . . . 15
2819, 26, 27sylanbrc 645 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
29 simprr2 1004 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
308adantr 451 . . . . . . . . . . . . . . . 16 Locally Locally t Locally t t
31 simplrl 736 . . . . . . . . . . . . . . . 16 Locally Locally t Locally t t
32 restabs 16896 . . . . . . . . . . . . . . . 16 t t t
3330, 20, 31, 32syl3anc 1182 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t t t t
34 simprr3 1005 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t t t
3533, 34eqeltrrd 2358 . . . . . . . . . . . . . 14 Locally Locally t Locally t t t
3628, 29, 35jca32 521 . . . . . . . . . . . . 13 Locally Locally t Locally t t t
3736ex 423 . . . . . . . . . . . 12 Locally Locally t Locally t t t
3818, 37syland 467 . . . . . . . . . . 11 Locally Locally t Locally t t t t
3938reximdv2 2652 . . . . . . . . . 10 Locally Locally t Locally t t t t
4014, 39mpd 14 . . . . . . . . 9 Locally Locally t Locally t
4140expr 598 . . . . . . . 8 Locally Locally t Locally t
4241rexlimdva 2667 . . . . . . 7 Locally Locally t Locally t
432, 42mpd 14 . . . . . 6 Locally Locally t
44433expb 1152 . . . . 5 Locally Locally t
4544ralrimivva 2635 . . . 4 Locally Locally t
46 islly 17194 . . . 4 Locally t
471, 45, 46sylanbrc 645 . . 3 Locally Locally Locally
4847ssriv 3184 . 2 Locally Locally Locally
49 llyrest 17211 . . . . 5 Locally t Locally
5049adantl 452 . . . 4 Locally t Locally
51 llytop 17198 . . . . . 6 Locally
5251ssriv 3184 . . . . 5 Locally
5352a1i 10 . . . 4 Locally
5450, 53restlly 17209 . . 3 Locally Locally Locally
5554trud 1314 . 2 Locally Locally Locally
5648, 55eqssi 3195 1 Locally Locally Locally
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   w3a 934   wtru 1307   wceq 1623   wcel 1684  wral 2543  wrex 2544   cin 3151   wss 3152  cpw 3625  (class class class)co 5858   ↾t crest 13325  ctop 16631  Locally clly 17190 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-lly 17192
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