Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  llyidm Structured version   Unicode version

Theorem llyidm 17551
 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 17535 . . . 4 Locally Locally
2 llyi 17537 . . . . . . 7 Locally Locally t Locally
3 simprr3 1007 . . . . . . . . 9 Locally Locally t Locally t Locally
4 simprl 733 . . . . . . . . . 10 Locally Locally t Locally
5 ssid 3367 . . . . . . . . . . 11
65a1i 11 . . . . . . . . . 10 Locally Locally t Locally
713ad2ant1 978 . . . . . . . . . . . 12 Locally Locally
87adantr 452 . . . . . . . . . . 11 Locally Locally t Locally
9 restopn2 17241 . . . . . . . . . . 11 t
108, 4, 9syl2anc 643 . . . . . . . . . 10 Locally Locally t Locally t
114, 6, 10mpbir2and 889 . . . . . . . . 9 Locally Locally t Locally t
12 simprr2 1006 . . . . . . . . 9 Locally Locally t Locally
13 llyi 17537 . . . . . . . . 9 t Locally t t t t
143, 11, 12, 13syl3anc 1184 . . . . . . . 8 Locally Locally t Locally t t t
15 restopn2 17241 . . . . . . . . . . . 12 t
168, 4, 15syl2anc 643 . . . . . . . . . . 11 Locally Locally t Locally t
17 simpl 444 . . . . . . . . . . 11
1816, 17syl6bi 220 . . . . . . . . . 10 Locally Locally t Locally t
19 simprl 733 . . . . . . . . . . . . 13 Locally Locally t Locally t t
20 simprr1 1005 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t
21 simprr1 1005 . . . . . . . . . . . . . . . 16 Locally Locally t Locally
2221adantr 452 . . . . . . . . . . . . . . 15 Locally Locally t Locally t t
2320, 22sstrd 3358 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
24 vex 2959 . . . . . . . . . . . . . . 15
2524elpw 3805 . . . . . . . . . . . . . 14
2623, 25sylibr 204 . . . . . . . . . . . . 13 Locally Locally t Locally t t
27 elin 3530 . . . . . . . . . . . . 13
2819, 26, 27sylanbrc 646 . . . . . . . . . . . 12 Locally Locally t Locally t t
29 simprr2 1006 . . . . . . . . . . . 12 Locally Locally t Locally t t
308adantr 452 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
31 simplrl 737 . . . . . . . . . . . . . 14 Locally Locally t Locally t t
32 restabs 17229 . . . . . . . . . . . . . 14 t t t
3330, 20, 31, 32syl3anc 1184 . . . . . . . . . . . . 13 Locally Locally t Locally t t t t t
34 simprr3 1007 . . . . . . . . . . . . 13 Locally Locally t Locally t t t t
3533, 34eqeltrrd 2511 . . . . . . . . . . . 12 Locally Locally t Locally t t t
3628, 29, 35jca32 522 . . . . . . . . . . 11 Locally Locally t Locally t t t
3736ex 424 . . . . . . . . . 10 Locally Locally t Locally t t t
3818, 37syland 468 . . . . . . . . 9 Locally Locally t Locally t t t t
3938reximdv2 2815 . . . . . . . 8 Locally Locally t Locally t t t t
4014, 39mpd 15 . . . . . . 7 Locally Locally t Locally t
412, 40rexlimddv 2834 . . . . . 6 Locally Locally t
42413expb 1154 . . . . 5 Locally Locally t
4342ralrimivva 2798 . . . 4 Locally Locally t
44 islly 17531 . . . 4 Locally t
451, 43, 44sylanbrc 646 . . 3 Locally Locally Locally
4645ssriv 3352 . 2 Locally Locally Locally
47 llyrest 17548 . . . . 5 Locally t Locally
4847adantl 453 . . . 4 Locally t Locally
49 llytop 17535 . . . . . 6 Locally
5049ssriv 3352 . . . . 5 Locally
5150a1i 11 . . . 4 Locally
5248, 51restlly 17546 . . 3 Locally Locally Locally
5352trud 1332 . 2 Locally Locally Locally
5446, 53eqssi 3364 1 Locally Locally Locally
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wtru 1325   wceq 1652   wcel 1725  wral 2705  wrex 2706   cin 3319   wss 3320  cpw 3799  (class class class)co 6081   ↾t crest 13648  ctop 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
 Copyright terms: Public domain W3C validator