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Theorem ispisys2 33146
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 33145 . 2 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3 dfss3 3970 . . . 4 ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦𝑆)
4 elex 3492 . . . . . . 7 (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ∈ V)
54adantr 481 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6 simpr 485 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7 eldifsn 4790 . . . . . . . . . 10 (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
86, 7sylib 217 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
98simpld 495 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
109elin1d 4198 . . . . . . 7 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
1110elpwid 4611 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥𝑆)
128simprd 496 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
139elin2d 4199 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14 elfir 9409 . . . . . 6 ((𝑆 ∈ V ∧ (𝑥𝑆𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → 𝑥 ∈ (fi‘𝑆))
155, 11, 12, 13, 14syl13anc 1372 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (fi‘𝑆))
16 elfi2 9408 . . . . . 6 (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥))
1716biimpa 477 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥)
18 simpr 485 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → 𝑦 = 𝑥)
1918eleq1d 2818 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → (𝑦𝑆 𝑥𝑆))
2015, 17, 19ralxfrd 5406 . . . 4 (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
213, 20bitrid 282 . . 3 (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
2221pm5.32i 575 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
232, 22bitri 274 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  {crab 3432  Vcvv 3474  cdif 3945  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628   cint 4950  cfv 6543  Fincfn 8938  ficfi 9404
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-fi 9405
This theorem is referenced by:  inelpisys  33147  sigapisys  33148  dynkin  33160
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