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Theorem llyidm 21101
 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 21085 . . . 4 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Top)
2 llyi 21087 . . . . . . 7 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))
3 simprr3 1104 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑗t 𝑢) ∈ Locally 𝐴)
4 simprl 790 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑗)
5 ssid 3587 . . . . . . . . . . 11 𝑢𝑢
65a1i 11 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑢)
713ad2ant1 1075 . . . . . . . . . . . 12 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → 𝑗 ∈ Top)
87adantr 480 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
9 restopn2 20791 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
108, 4, 9syl2anc 691 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
114, 6, 10mpbir2and 959 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
12 simprr2 1103 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑦𝑢)
13 llyi 21087 . . . . . . . . 9 (((𝑗t 𝑢) ∈ Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
143, 11, 12, 13syl3anc 1318 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
15 restopn2 20791 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
168, 4, 15syl2anc 691 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
17 simpl 472 . . . . . . . . . . 11 ((𝑣𝑗𝑣𝑢) → 𝑣𝑗)
1816, 17syl6bi 242 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) → 𝑣𝑗))
19 simprl 790 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑗)
20 simprr1 1102 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
21 simprr1 1102 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑥)
2221adantr 480 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
2320, 22sstrd 3578 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
24 selpw 4115 . . . . . . . . . . . . . 14 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2523, 24sylibr 223 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2619, 25elind 3760 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝑗 ∩ 𝒫 𝑥))
27 simprr2 1103 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
288adantr 480 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
29 simplrl 796 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
30 restabs 20779 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3128, 20, 29, 30syl3anc 1318 . . . . . . . . . . . . 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 1104 . . . . . . . . . . . . 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 2689 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
3426, 27, 33jca32 556 . . . . . . . . . . 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 449 . . . . . . . . . 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 497 . . . . . . . . 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 2997 . . . . . . . 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 3017 . . . . . 6 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
40393expb 1258 . . . . 5 ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
4140ralrimivva 2954 . . . 4 (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
42 islly 21081 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
431, 41, 42sylanbrc 695 . . 3 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Locally 𝐴)
4443ssriv 3572 . 2 Locally Locally 𝐴 ⊆ Locally 𝐴
45 llyrest 21098 . . . . 5 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
4645adantl 481 . . . 4 ((⊤ ∧ (𝑗 ∈ Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
47 llytop 21085 . . . . . 6 (𝑗 ∈ Locally 𝐴𝑗 ∈ Top)
4847ssriv 3572 . . . . 5 Locally 𝐴 ⊆ Top
4948a1i 11 . . . 4 (⊤ → Locally 𝐴 ⊆ Top)
5046, 49restlly 21096 . . 3 (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
5150trud 1484 . 2 Locally 𝐴 ⊆ Locally Locally 𝐴
5244, 51eqssi 3584 1 Locally Locally 𝐴 = Locally 𝐴
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Locally clly 21077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-lly 21079 This theorem is referenced by: (None)
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