Theorem isfil2 23207

Description: Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
isfil2	⊢ (𝐹 ∈ (Fil‘𝑋) ↔ ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦

Proof of Theorem isfil2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		filsspw 23202	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2		0nelfil 23200	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3		filtop 23206	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	1, 2, 3	3jca 1128	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹))
5		elpwi 4567	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
6		filss 23204	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
7	6	3exp2 1354	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑦 ∈ 𝐹 → (𝑥 ⊆ 𝑋 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))))
8	7	com23 86	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑦 ∈ 𝐹 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))))
9	8	imp 407	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑦 ∈ 𝐹 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹)))
10	9	rexlimdv 3150	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
11	5, 10	sylan2 593	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
12	11	ralrimiva 3143	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
13		filin 23205	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
14	13	3expb 1120	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
15	14	ralrimivva 3197	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹)
16	4, 12, 15	3jca 1128	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹))
17		simp11 1203	. . . 4 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
18		simp13 1205	. . . . . 6 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → 𝑋 ∈ 𝐹)
19	18	ne0d 4295	. . . . 5 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → 𝐹 ≠ ∅)
20		simp12 1204	. . . . . 6 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → ¬ ∅ ∈ 𝐹)
21		df-nel 3050	. . . . . 6 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
22	20, 21	sylibr 233	. . . . 5 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → ∅ ∉ 𝐹)
23		ssid 3966	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
24		sseq1 3969	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
25	24	rspcev 3581	. . . . . . . . 9 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
26	23, 25	mpan2 689	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐹 → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
27	26	ralimi 3086	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹 → ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
28	27	ralimi 3086	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
29	28	3ad2ant3 1135	. . . . 5 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
30	19, 22, 29	3jca 1128	. . . 4 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))
31		isfbas2 23186	. . . . 5 ⊢ (𝑋 ∈ 𝐹 → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
32	18, 31	syl 17	. . . 4 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
33	17, 30, 32	mpbir2and 711	. . 3 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
34		n0 4306	. . . . . . . 8 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹 ∩ 𝒫 𝑥))
35		elin 3926	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ 𝐹 ∧ 𝑦 ∈ 𝒫 𝑥))
36		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑥 → 𝑦 ⊆ 𝑥)
37	36	anim2i 617	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ∈ 𝒫 𝑥) → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
38	35, 37	sylbi 216	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
39	38	eximi 1837	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
40	34, 39	sylbi 216	. . . . . . 7 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
41		df-rex 3074	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 ↔ ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
42	40, 41	sylibr 233	. . . . . 6 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)
43	42	imim1i 63	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) → ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))
44	43	ralimi 3086	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))
45	44	3ad2ant2 1134	. . 3 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))
46		isfil 23198	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)))
47	33, 45, 46	sylanbrc 583	. 2 ⊢ (((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
48	16, 47	impbii 208	1 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ‘cfv 6496  fBascfbas 20784  Filcfil 23196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-fbas 20793  df-fil 23197

This theorem is referenced by:  isfild  23209  infil  23214  neifil  23231  trfil2  23238