Theorem isfil2 21893

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 21888	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2		0nelfil 21886	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3		filtop 21892	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	1, 2, 3	3jca 1151	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹))
5		elpwi 4372	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
6		filss 21890	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
7	6	3exp2 1456	. . . . . . . 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 395	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑦 ∈ 𝐹 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹)))
10	9	rexlimdv 3229	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
11	5, 10	sylan2 582	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
12	11	ralrimiva 3165	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
13		filin 21891	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
14	13	3expb 1142	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
15	14	ralrimivva 3170	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹)
16	4, 12, 15	3jca 1151	. 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 1253	. . . 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 1255	. . . . . 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 4134	. . . . 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 1254	. . . . . 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 3093	. . . . . 6 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
22	20, 21	sylibr 225	. . . . 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 3831	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
24		sseq1 3834	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
25	24	rspcev 3513	. . . . . . . . 9 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
26	23, 25	mpan2 674	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐹 → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
27	26	ralimi 3151	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹 → ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
28	27	ralimi 3151	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
29	28	3ad2ant3 1158	. . . . 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 1151	. . . 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 21872	. . . . 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 695	. . 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 4143	. . . . . . . 8 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹 ∩ 𝒫 𝑥))
35		elin 4006	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ 𝐹 ∧ 𝑦 ∈ 𝒫 𝑥))
36		elpwi 4372	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑥 → 𝑦 ⊆ 𝑥)
37	36	anim2i 605	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ∈ 𝒫 𝑥) → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
38	35, 37	sylbi 208	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
39	38	eximi 1919	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
40	34, 39	sylbi 208	. . . . . . 7 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
41		df-rex 3113	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 ↔ ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
42	40, 41	sylibr 225	. . . . . 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 3151	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))
45	44	3ad2ant2 1157	. . 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 21884	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)))
47	33, 45, 46	sylanbrc 574	. 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 200	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 197   ∧ wa 384   ∧ w3a 1100  ∃wex 1859   ∈ wcel 2157   ≠ wne 2989   ∉ wnel 3092  ∀wral 3107  ∃wrex 3108   ∩ cin 3779   ⊆ wss 3780  ∅c0 4127  𝒫 cpw 4362  ‘cfv 6110  fBascfbas 19961  Filcfil 21882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fv 6118  df-fbas 19970  df-fil 21883

This theorem is referenced by:  isfild  21895  infil  21900  neifil  21917  trfil2  21924