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Theorem llyidm 21800
 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 21784 . . . 4 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Top)
2 llyi 21786 . . . . . . 7 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))
3 simprr3 1203 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑗t 𝑢) ∈ Locally 𝐴)
4 simprl 758 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑗)
5 ssidd 3880 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑢)
613ad2ant1 1113 . . . . . . . . . . . 12 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → 𝑗 ∈ Top)
76adantr 473 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
8 restopn2 21489 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
97, 4, 8syl2anc 576 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
104, 5, 9mpbir2and 700 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
11 simprr2 1202 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑦𝑢)
12 llyi 21786 . . . . . . . . 9 (((𝑗t 𝑢) ∈ Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
133, 10, 11, 12syl3anc 1351 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
14 restopn2 21489 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
157, 4, 14syl2anc 576 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
16 simpl 475 . . . . . . . . . . 11 ((𝑣𝑗𝑣𝑢) → 𝑣𝑗)
1715, 16syl6bi 245 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) → 𝑣𝑗))
18 simprl 758 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑗)
19 simprr1 1201 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
20 simprr1 1201 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑥)
2120adantr 473 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
2219, 21sstrd 3868 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
23 selpw 4429 . . . . . . . . . . . . . 14 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2422, 23sylibr 226 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2518, 24elind 4059 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝑗 ∩ 𝒫 𝑥))
26 simprr2 1202 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
277adantr 473 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
28 simplrl 764 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
29 restabs 21477 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3027, 19, 28, 29syl3anc 1351 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
31 simprr3 1203 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)
3230, 31eqeltrrd 2867 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
3325, 26, 32jca32 508 . . . . . . . . . . 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)))
3433ex 405 . . . . . . . . . 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))))
3517, 34syland 593 . . . . . . . . 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))))
3635reximdv2 3216 . . . . . . . 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)))
3713, 36mpd 15 . . . . . . 7 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
382, 37rexlimddv 3236 . . . . . 6 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
39383expb 1100 . . . . 5 ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
4039ralrimivva 3141 . . . 4 (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
41 islly 21780 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
421, 40, 41sylanbrc 575 . . 3 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Locally 𝐴)
4342ssriv 3862 . 2 Locally Locally 𝐴 ⊆ Locally 𝐴
44 llyrest 21797 . . . . 5 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
4544adantl 474 . . . 4 ((⊤ ∧ (𝑗 ∈ Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
46 llytop 21784 . . . . . 6 (𝑗 ∈ Locally 𝐴𝑗 ∈ Top)
4746ssriv 3862 . . . . 5 Locally 𝐴 ⊆ Top
4847a1i 11 . . . 4 (⊤ → Locally 𝐴 ⊆ Top)
4945, 48restlly 21795 . . 3 (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
5049mptru 1514 . 2 Locally 𝐴 ⊆ Locally Locally 𝐴
5143, 50eqssi 3874 1 Locally Locally 𝐴 = Locally 𝐴
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507  ⊤wtru 1508   ∈ wcel 2050  ∀wral 3088  ∃wrex 3089   ∩ cin 3828   ⊆ wss 3829  𝒫 cpw 4422  (class class class)co 6976   ↾t crest 16550  Topctop 21205  Locally clly 21776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-oadd 7909  df-er 8089  df-en 8307  df-fin 8310  df-fi 8670  df-rest 16552  df-topgen 16573  df-top 21206  df-topon 21223  df-bases 21258  df-lly 21778 This theorem is referenced by: (None)
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