Theorem ispisys2 34310

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 34309	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3		dfss3 3922	. . . 4 ⊢ ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆)
4		elex 3461	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → 𝑆 ∈ V)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6		simpr 484	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7		eldifsn 4742	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
8	6, 7	sylib 218	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
9	8	simpld 494	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
10	9	elin1d 4156	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
11	10	elpwid 4563	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ⊆ 𝑆)
12	8	simprd 495	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
13	9	elin2d 4157	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14		elfir 9318	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ (𝑥 ⊆ 𝑆 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → ∩ 𝑥 ∈ (fi‘𝑆))
15	5, 11, 12, 13, 14	syl13anc 1374	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → ∩ 𝑥 ∈ (fi‘𝑆))
16		elfi2 9317	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥))
17	16	biimpa 476	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥)
18		simpr 484	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
19	18	eleq1d 2821	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → (𝑦 ∈ 𝑆 ↔ ∩ 𝑥 ∈ 𝑆))
20	15, 17, 19	ralxfrd 5353	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
21	3, 20	bitrid 283	. . 3 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
22	21	pm5.32i 574	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
23	2, 22	bitri 275	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ∩ cint 4902  ‘cfv 6492  Fincfn 8883  ficfi 9313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-fi 9314

This theorem is referenced by:  inelpisys  34311  sigapisys  34312  dynkin  34324