Theorem ispisys2 34344

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 34343	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3		dfss3 3911	. . . 4 ⊢ ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆)
4		elex 3453	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → 𝑆 ∈ V)
5	4	adantr 481	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6		eldifsn 4726	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
7	6	bilani 505	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
8	7	simpld 495	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
9	8	elin1d 4140	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
10	9	elpwid 4545	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ⊆ 𝑆)
11	7	simprd 496	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
12	8	elin2d 4141	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
13		elfir 9325	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ (𝑥 ⊆ 𝑆 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → ∩ 𝑥 ∈ (fi‘𝑆))
14	5, 10, 11, 12, 13	syl13anc 1380	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → ∩ 𝑥 ∈ (fi‘𝑆))
15		elfi2 9324	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥))
16	15	biimpa 477	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥)
17		simpr 485	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
18	17	eleq1d 2825	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → (𝑦 ∈ 𝑆 ↔ ∩ 𝑥 ∈ 𝑆))
19	14, 16, 18	ralxfrd 5344	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
20	3, 19	bitrid 284	. . 3 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
21	20	pm5.32i 579	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
22	2, 21	bitri 276	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  {crab 3392  Vcvv 3432   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  {csn 4562  ∩ cint 4884  ‘cfv 6492  Fincfn 8890  ficfi 9320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-fi 9321

This theorem is referenced by:  inelpisys  34345  sigapisys  34346  dynkin  34358