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Theorem llyidm 21214
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 21198 . . . 4 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Top)
2 llyi 21200 . . . . . . 7 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))
3 simprr3 1109 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑗t 𝑢) ∈ Locally 𝐴)
4 simprl 793 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑗)
5 ssid 3608 . . . . . . . . . . 11 𝑢𝑢
65a1i 11 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑢)
713ad2ant1 1080 . . . . . . . . . . . 12 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → 𝑗 ∈ Top)
87adantr 481 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
9 restopn2 20904 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
108, 4, 9syl2anc 692 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
114, 6, 10mpbir2and 956 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
12 simprr2 1108 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑦𝑢)
13 llyi 21200 . . . . . . . . 9 (((𝑗t 𝑢) ∈ Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
143, 11, 12, 13syl3anc 1323 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
15 restopn2 20904 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
168, 4, 15syl2anc 692 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
17 simpl 473 . . . . . . . . . . 11 ((𝑣𝑗𝑣𝑢) → 𝑣𝑗)
1816, 17syl6bi 243 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) → 𝑣𝑗))
19 simprl 793 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑗)
20 simprr1 1107 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
21 simprr1 1107 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑥)
2221adantr 481 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
2320, 22sstrd 3597 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
24 selpw 4142 . . . . . . . . . . . . . 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 3781 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝑗 ∩ 𝒫 𝑥))
27 simprr2 1108 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
288adantr 481 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
29 simplrl 799 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
30 restabs 20892 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3128, 20, 29, 30syl3anc 1323 . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . 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 2699 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
3426, 27, 33jca32 557 . . . . . . . . . . 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 450 . . . . . . . . . 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 498 . . . . . . . . 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 3009 . . . . . . . 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 3029 . . . . . 6 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
40393expb 1263 . . . . 5 ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
4140ralrimivva 2966 . . . 4 (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
42 islly 21194 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
431, 41, 42sylanbrc 697 . . 3 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Locally 𝐴)
4443ssriv 3591 . 2 Locally Locally 𝐴 ⊆ Locally 𝐴
45 llyrest 21211 . . . . 5 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
4645adantl 482 . . . 4 ((⊤ ∧ (𝑗 ∈ Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
47 llytop 21198 . . . . . 6 (𝑗 ∈ Locally 𝐴𝑗 ∈ Top)
4847ssriv 3591 . . . . 5 Locally 𝐴 ⊆ Top
4948a1i 11 . . . 4 (⊤ → Locally 𝐴 ⊆ Top)
5046, 49restlly 21209 . . 3 (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
5150trud 1490 . 2 Locally 𝐴 ⊆ Locally Locally 𝐴
5244, 51eqssi 3603 1 Locally Locally 𝐴 = Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1987  wral 2907  wrex 2908  cin 3558  wss 3559  𝒫 cpw 4135  (class class class)co 6610  t crest 16013  Topctop 20630  Locally clly 21190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-en 7908  df-fin 7911  df-fi 8269  df-rest 16015  df-topgen 16036  df-top 20631  df-topon 20648  df-bases 20674  df-lly 21192
This theorem is referenced by: (None)
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