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Theorem ispisys2 32816
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 32815 . 2 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3 dfss3 3936 . . . 4 ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦𝑆)
4 elex 3465 . . . . . . 7 (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ∈ V)
54adantr 482 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6 simpr 486 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7 eldifsn 4751 . . . . . . . . . 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 496 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
109elin1d 4162 . . . . . . 7 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
1110elpwid 4573 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥𝑆)
128simprd 497 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
139elin2d 4163 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14 elfir 9359 . . . . . 6 ((𝑆 ∈ V ∧ (𝑥𝑆𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → 𝑥 ∈ (fi‘𝑆))
155, 11, 12, 13, 14syl13anc 1373 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (fi‘𝑆))
16 elfi2 9358 . . . . . 6 (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥))
1716biimpa 478 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥)
18 simpr 486 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → 𝑦 = 𝑥)
1918eleq1d 2819 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → (𝑦𝑆 𝑥𝑆))
2015, 17, 19ralxfrd 5367 . . . 4 (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
213, 20bitrid 283 . . 3 (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
2221pm5.32i 576 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
232, 22bitri 275 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  {crab 3406  Vcvv 3447  cdif 3911  cin 3913  wss 3914  c0 4286  𝒫 cpw 4564  {csn 4590   cint 4911  cfv 6500  Fincfn 8889  ficfi 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-fi 9355
This theorem is referenced by:  inelpisys  32817  sigapisys  32818  dynkin  32830
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