Theorem ispisys2 32121

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	⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))

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.

Step	Hyp	Ref	Expression
1		ispisys.p	. . 3 ⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
2	1	ispisys 32120	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3		dfss3 3909	. . . 4 ⊢ ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆)
4		elex 3450	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → 𝑆 ∈ V)
5	4	adantr 481	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6		simpr 485	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7		eldifsn 4720	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
8	6, 7	sylib 217	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
9	8	simpld 495	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
10	9	elin1d 4132	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
11	10	elpwid 4544	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ⊆ 𝑆)
12	8	simprd 496	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
13	9	elin2d 4133	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14		elfir 9174	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ (𝑥 ⊆ 𝑆 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → ∩ 𝑥 ∈ (fi‘𝑆))
15	5, 11, 12, 13, 14	syl13anc 1371	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → ∩ 𝑥 ∈ (fi‘𝑆))
16		elfi2 9173	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥))
17	16	biimpa 477	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥)
18		simpr 485	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
19	18	eleq1d 2823	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → (𝑦 ∈ 𝑆 ↔ ∩ 𝑥 ∈ 𝑆))
20	15, 17, 19	ralxfrd 5331	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
21	3, 20	syl5bb 283	. . 3 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
22	21	pm5.32i 575	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
23	2, 22	bitri 274	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∩ cint 4879  ‘cfv 6433  Fincfn 8733  ficfi 9169

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-fi 9170

This theorem is referenced by:  inelpisys  32122  sigapisys  32123  dynkin  32135