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Theorem ispisys2 30009
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 30008 . 2 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3 dfss3 3574 . . . 4 ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦𝑆)
4 elex 3198 . . . . . . 7 (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ∈ V)
54adantr 481 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6 simpr 477 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7 eldifsn 4289 . . . . . . . . . 10 (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
86, 7sylib 208 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
98simpld 475 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
109elin1d 3782 . . . . . . 7 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
1110elpwid 4143 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥𝑆)
128simprd 479 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
139elin2d 3783 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14 elfir 8268 . . . . . 6 ((𝑆 ∈ V ∧ (𝑥𝑆𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → 𝑥 ∈ (fi‘𝑆))
155, 11, 12, 13, 14syl13anc 1325 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (fi‘𝑆))
16 elfi2 8267 . . . . . 6 (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥))
1716biimpa 501 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = 𝑥)
18 simpr 477 . . . . . 6 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → 𝑦 = 𝑥)
1918eleq1d 2683 . . . . 5 ((𝑆 ∈ 𝒫 𝒫 𝑂𝑦 = 𝑥) → (𝑦𝑆 𝑥𝑆))
2015, 17, 19ralxfrd 4841 . . . 4 (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
213, 20syl5bb 272 . . 3 (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
2221pm5.32i 668 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
232, 22bitri 264 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) 𝑥𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3553  cin 3555  wss 3556  c0 3893  𝒫 cpw 4132  {csn 4150   cint 4442  cfv 5849  Fincfn 7902  ficfi 8263
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-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-fi 8264
This theorem is referenced by:  inelpisys  30010  sigapisys  30011  dynkin  30023
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