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Theorem llyidm 21511
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 21495 . . . 4 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Top)
2 llyi 21497 . . . . . . 7 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))
3 simprr3 1276 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑗t 𝑢) ∈ Locally 𝐴)
4 simprl 754 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑗)
5 ssid 3773 . . . . . . . . . . 11 𝑢𝑢
65a1i 11 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑢)
713ad2ant1 1127 . . . . . . . . . . . 12 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → 𝑗 ∈ Top)
87adantr 466 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
9 restopn2 21201 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
108, 4, 9syl2anc 573 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
114, 6, 10mpbir2and 692 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
12 simprr2 1274 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑦𝑢)
13 llyi 21497 . . . . . . . . 9 (((𝑗t 𝑢) ∈ Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
143, 11, 12, 13syl3anc 1476 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
15 restopn2 21201 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
168, 4, 15syl2anc 573 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
17 simpl 468 . . . . . . . . . . 11 ((𝑣𝑗𝑣𝑢) → 𝑣𝑗)
1816, 17syl6bi 243 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) → 𝑣𝑗))
19 simprl 754 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑗)
20 simprr1 1272 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
21 simprr1 1272 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑥)
2221adantr 466 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
2320, 22sstrd 3762 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
24 selpw 4305 . . . . . . . . . . . . . 14 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2523, 24sylibr 224 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2619, 25elind 3949 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝑗 ∩ 𝒫 𝑥))
27 simprr2 1274 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
288adantr 466 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
29 simplrl 762 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
30 restabs 21189 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3128, 20, 29, 30syl3anc 1476 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
32 simprr3 1276 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)
3331, 32eqeltrrd 2851 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
3426, 27, 33jca32 505 . . . . . . . . . . 11 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
3534ex 397 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ((𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))))
3618, 35syland 590 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ((𝑣 ∈ (𝑗t 𝑢) ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))))
3736reximdv2 3162 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
3814, 37mpd 15 . . . . . . 7 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
392, 38rexlimddv 3183 . . . . . 6 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
40393expb 1113 . . . . 5 ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
4140ralrimivva 3120 . . . 4 (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
42 islly 21491 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
431, 41, 42sylanbrc 572 . . 3 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Locally 𝐴)
4443ssriv 3756 . 2 Locally Locally 𝐴 ⊆ Locally 𝐴
45 llyrest 21508 . . . . 5 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
4645adantl 467 . . . 4 ((⊤ ∧ (𝑗 ∈ Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
47 llytop 21495 . . . . . 6 (𝑗 ∈ Locally 𝐴𝑗 ∈ Top)
4847ssriv 3756 . . . . 5 Locally 𝐴 ⊆ Top
4948a1i 11 . . . 4 (⊤ → Locally 𝐴 ⊆ Top)
5046, 49restlly 21506 . . 3 (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
5150trud 1641 . 2 Locally 𝐴 ⊆ Locally Locally 𝐴
5244, 51eqssi 3768 1 Locally Locally 𝐴 = Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wtru 1632  wcel 2145  wral 3061  wrex 3062  cin 3722  wss 3723  𝒫 cpw 4298  (class class class)co 6795  t crest 16288  Topctop 20917  Locally clly 21487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-oadd 7720  df-er 7899  df-en 8113  df-fin 8116  df-fi 8476  df-rest 16290  df-topgen 16311  df-top 20918  df-topon 20935  df-bases 20970  df-lly 21489
This theorem is referenced by: (None)
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