Theorem isfil2 24013

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 24008	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2		0nelfil 24006	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3		filtop 24012	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	1, 2, 3	3jca 1146	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹))
5		elpwi 4569	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
6		filss 24010	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
7	6	3exp2 1373	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑦 ∈ 𝐹 → (𝑥 ⊆ 𝑋 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))))
8	7	com23 87	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑦 ∈ 𝐹 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))))
9	8	imp 411	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑦 ∈ 𝐹 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹)))
10	9	rexlimdv 3164	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
11	5, 10	sylan2 604	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
12	11	ralrimiva 3157	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹))
13		filin 24011	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
14	13	3expb 1138	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
15	14	ralrimivva 3208	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹)
16	4, 12, 15	3jca 1146	. 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 1222	. . . 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 1224	. . . . . 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 1223	. . . . . 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 3065	. . . . . 6 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
22	20, 21	sylibr 237	. . . . 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 3959	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
24		sseq1 3962	. . . . . . . . . 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 703	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐹 → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
27	26	ralimi 3102	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹 → ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
28	27	ralimi 3102	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹 → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
29	28	3ad2ant3 1153	. . . . 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 1146	. . . 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 23992	. . . . 5 ⊢ (𝑋 ∈ 𝐹 → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
32	18, 31	syl 18	. . . 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 725	. . 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 4307	. . . . . . . 8 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹 ∩ 𝒫 𝑥))
35		elin 3921	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ 𝐹 ∧ 𝑦 ∈ 𝒫 𝑥))
36		elpwi 4569	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑥 → 𝑦 ⊆ 𝑥)
37	36	anim2i 628	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ∈ 𝒫 𝑥) → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
38	35, 37	sylbi 220	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) → (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
39	38	eximi 1865	. . . . . . . 8 ⊢ (∃𝑦 𝑦 ∈ (𝐹 ∩ 𝒫 𝑥) → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
40	34, 39	sylbi 220	. . . . . . 7 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
41		df-rex 3090	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 ↔ ∃𝑦(𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥))
42	40, 41	sylibr 237	. . . . . 6 ⊢ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)
43	42	imim1i 64	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) → ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))
44	43	ralimi 3102	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))
45	44	3ad2ant2 1152	. . 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 24004	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)))
47	33, 45, 46	sylanbrc 594	. 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 212	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 209   ∧ wa 400   ∧ w3a 1103  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958   ∉ wnel 3064  ∀wral 3079  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  ‘cfv 6536  fBascfbas 21510  Filcfil 24002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-fbas 21519  df-fil 24003

This theorem is referenced by:  isfild  24015  infil  24020  neifil  24037  trfil2  24044