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Theorem ispisys2 31653
 Description: The property of being a pi-system, expanded version. Pi-systems are closed under finite intersections. (Contributed by Thierry Arnoux, 13-Jun-2020.)
Hypothesis
Ref Expression
ispisys.p 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
Assertion
Ref Expression
ispisys2 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
Distinct variable groups:   𝑂,𝑠,𝑥   𝑆,𝑠,𝑥
Allowed substitution hints:   𝑃(𝑥,𝑠)

Proof of Theorem ispisys2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ispisys.p . . 3 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
21ispisys 31652 . 2 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3 dfss3 3882 . . . 4 ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦𝑆)
4 elex 3428 . . . . . . 7 (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ∈ V)
54adantr 484 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6 simpr 488 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7 eldifsn 4680 . . . . . . . . . 10 (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
86, 7sylib 221 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
98simpld 498 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
109elin1d 4105 . . . . . . 7 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
1110elpwid 4508 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥𝑆)
128simprd 499 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
139elin2d 4106 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14 elfir 8925 . . . . . 6 ((𝑆 ∈ V ∧ (𝑥𝑆𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → 𝑥 ∈ (fi‘𝑆))
155, 11, 12, 13, 14syl13anc 1369 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (fi‘𝑆))
16 elfi2 8924 . . . . . 6 (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥))
1716biimpa 480 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥)
18 simpr 488 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → 𝑦 = 𝑥)
1918eleq1d 2836 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → (𝑦𝑆 𝑥𝑆))
2015, 17, 19ralxfrd 5281 . . . 4 (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
213, 20syl5bb 286 . . 3 (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
2221pm5.32i 578 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
232, 22bitri 278 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ∩ cint 4841  ‘cfv 6340  Fincfn 8540  ficfi 8920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-fi 8921 This theorem is referenced by:  inelpisys  31654  sigapisys  31655  dynkin  31667
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