Theorem ispisys2 31412

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 31411	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
3		dfss3 3956	. . . 4 ⊢ ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆)
4		elex 3512	. . . . . . 7 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → 𝑆 ∈ V)
5	4	adantr 483	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑆 ∈ V)
6		simpr 487	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}))
7		eldifsn 4719	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
8	6, 7	sylib 220	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → (𝑥 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑥 ≠ ∅))
9	8	simpld 497	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ (𝒫 𝑆 ∩ Fin))
10	9	elin1d 4175	. . . . . . 7 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ 𝒫 𝑆)
11	10	elpwid 4550	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ⊆ 𝑆)
12	8	simprd 498	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ≠ ∅)
13	9	elin2d 4176	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → 𝑥 ∈ Fin)
14		elfir 8879	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ (𝑥 ⊆ 𝑆 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)) → ∩ 𝑥 ∈ (fi‘𝑆))
15	5, 11, 12, 13, 14	syl13anc 1368	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})) → ∩ 𝑥 ∈ (fi‘𝑆))
16		elfi2 8878	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (𝑦 ∈ (fi‘𝑆) ↔ ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥))
17	16	biimpa 479	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 ∈ (fi‘𝑆)) → ∃𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})𝑦 = ∩ 𝑥)
18		simpr 487	. . . . . 6 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
19	18	eleq1d 2897	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ 𝑦 = ∩ 𝑥) → (𝑦 ∈ 𝑆 ↔ ∩ 𝑥 ∈ 𝑆))
20	15, 17, 19	ralxfrd 5309	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → (∀𝑦 ∈ (fi‘𝑆)𝑦 ∈ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
21	3, 20	syl5bb 285	. . 3 ⊢ (𝑆 ∈ 𝒫 𝒫 𝑂 → ((fi‘𝑆) ⊆ 𝑆 ↔ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
22	21	pm5.32i 577	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))
23	2, 22	bitri 277	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∀𝑥 ∈ ((𝒫 𝑆 ∩ Fin) ∖ {∅})∩ 𝑥 ∈ 𝑆))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  ∩ cint 4876  ‘cfv 6355  Fincfn 8509  ficfi 8874

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-fi 8875

This theorem is referenced by:  inelpisys  31413  sigapisys  31414  dynkin  31426