Theorem ispisys2 34460

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 34459	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3		dfss3 3928	. . . 4 ⊢ ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆)
4		elex 3478	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → 𝑆 ∈ V)
5	4	adantr 485	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6		eldifsn 4749	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
7	6	bilani 509	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
8	7	simpld 499	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
9	8	elin1d 4159	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
10	9	elpwid 4567	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ⊆ 𝑆)
11	7	simprd 500	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
12	8	elin2d 4160	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
13		elfir 9363	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ (𝑥 ⊆ 𝑆 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → ∩ 𝑥 ∈ (fi‘𝑆))
14	5, 10, 11, 12, 13	syl13anc 1395	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → ∩ 𝑥 ∈ (fi‘𝑆))
15		elfi2 9362	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥))
16	15	biimpa 481	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥)
17		simpr 489	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
18	17	eleq1d 2850	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → (𝑦 ∈ 𝑆 ↔ ∩ 𝑥 ∈ 𝑆))
19	14, 16, 18	ralxfrd 5370	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
20	3, 19	bitrid 286	. . 3 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
21	20	pm5.32i 584	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
22	2, 21	bitri 278	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  {csn 4585  ∩ cint 4908  ‘cfv 6525  Fincfn 8931  ficfi 9358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-fi 9359

This theorem is referenced by:  inelpisys  34461  sigapisys  34462  dynkin  34474