Theorem fbssfi 23845

Description: A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Assertion

Ref	Expression
fbssfi	⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)

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

Proof of Theorem fbssfi

Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffi2 9463	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (fi‘𝐹) = ∩ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)})
2		sseq2 4010	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 ∩ 𝑣) → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ (𝑢 ∩ 𝑣)))
3	2	rexbidv 3179	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 ∩ 𝑣) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑢 ∩ 𝑣)))
4		inss1 4237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
5		simp1r 1199	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → 𝑢 ∈ 𝒫 ∪ 𝐹)
6	5	elpwid 4609	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → 𝑢 ⊆ ∪ 𝐹)
7	4, 6	sstrid 3995	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐹)
8		vex 3484	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
9	8	inex1 5317	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝑣) ∈ V
10	9	elpw 4604	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∩ 𝑣) ∈ 𝒫 ∪ 𝐹 ↔ (𝑢 ∩ 𝑣) ⊆ ∪ 𝐹)
11	7, 10	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑢 ∩ 𝑣) ∈ 𝒫 ∪ 𝐹)
12		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
13		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) → 𝑦 ∈ 𝐹)
14		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣) → 𝑧 ∈ 𝐹)
15		fbasssin 23844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
16	12, 13, 14, 15	syl3an 1161	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
17		ss2in 4245	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ⊆ 𝑢 ∧ 𝑧 ⊆ 𝑣) → (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣))
18	17	ad2ant2l 746	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣))
19	18	3adant1 1131	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣))
20		sstr 3992	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣)) → 𝑥 ⊆ (𝑢 ∩ 𝑣))
21	20	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣) → (𝑥 ⊆ (𝑦 ∩ 𝑧) → 𝑥 ⊆ (𝑢 ∩ 𝑣)))
22	19, 21	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑥 ⊆ (𝑦 ∩ 𝑧) → 𝑥 ⊆ (𝑢 ∩ 𝑣)))
23	22	reximdv 3170	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑢 ∩ 𝑣)))
24	16, 23	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑢 ∩ 𝑣))
25	3, 11, 24	elrabd 3694	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
26	25	3expa 1119	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
27	26	rexlimdvaa 3156	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢)) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣 → (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
28	27	ralrimivw 3150	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢)) → ∀𝑣 ∈ 𝒫 ∪ 𝐹(∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣 → (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
29		sseq2 4010	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑣 → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ 𝑣))
30	29	rexbidv 3179	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑣 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑣))
31		sseq1 4009	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝑣 ↔ 𝑧 ⊆ 𝑣))
32	31	cbvrexvw 3238	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑣 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣)
33	30, 32	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣))
34	33	ralrab 3699	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ ∀𝑣 ∈ 𝒫 ∪ 𝐹(∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣 → (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
35	28, 34	sylibr 234	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢)) → ∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
36	35	rexlimdvaa 3156	. . . . . . . . 9 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) → (∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢 → ∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
37	36	ralrimiva 3146	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑢 ∈ 𝒫 ∪ 𝐹(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢 → ∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
38		sseq2 4010	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑢 → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ 𝑢))
39	38	rexbidv 3179	. . . . . . . . . 10 ⊢ (𝑡 = 𝑢 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑢))
40		sseq1 4009	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑢 ↔ 𝑦 ⊆ 𝑢))
41	40	cbvrexvw 3238	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑢 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢)
42	39, 41	bitrdi 287	. . . . . . . . 9 ⊢ (𝑡 = 𝑢 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
43	42	ralrab 3699	. . . . . . . 8 ⊢ (∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ ∀𝑢 ∈ 𝒫 ∪ 𝐹(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢 → ∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
44	37, 43	sylibr 234	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
45		pwuni 4945	. . . . . . . 8 ⊢ 𝐹 ⊆ 𝒫 ∪ 𝐹
46		ssid 4006	. . . . . . . . . 10 ⊢ 𝑡 ⊆ 𝑡
47		sseq1 4009	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡))
48	47	rspcev 3622	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)
49	46, 48	mpan2 691	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)
50	49	rgen 3063	. . . . . . . 8 ⊢ ∀𝑡 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡
51		ssrab 4073	. . . . . . . 8 ⊢ (𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ (𝐹 ⊆ 𝒫 ∪ 𝐹 ∧ ∀𝑡 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
52	45, 50, 51	mpbir2an 711	. . . . . . 7 ⊢ 𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}
53	44, 52	jctil 519	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∧ ∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
54		uniexg 7760	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∪ 𝐹 ∈ V)
55		pwexg 5378	. . . . . . 7 ⊢ (∪ 𝐹 ∈ V → 𝒫 ∪ 𝐹 ∈ V)
56		rabexg 5337	. . . . . . 7 ⊢ (𝒫 ∪ 𝐹 ∈ V → {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ V)
57		sseq2 4010	. . . . . . . . 9 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → (𝐹 ⊆ 𝑧 ↔ 𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
58		eleq2 2830	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → ((𝑢 ∩ 𝑣) ∈ 𝑧 ↔ (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
59	58	raleqbi1dv 3338	. . . . . . . . . 10 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → (∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧 ↔ ∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
60	59	raleqbi1dv 3338	. . . . . . . . 9 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → (∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧 ↔ ∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
61	57, 60	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → ((𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧) ↔ (𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∧ ∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})))
62	61	elabg 3676	. . . . . . 7 ⊢ ({𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ V → ({𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)} ↔ (𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∧ ∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})))
63	54, 55, 56, 62	4syl 19	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → ({𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)} ↔ (𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∧ ∀𝑢 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}∀𝑣 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})))
64	53, 63	mpbird 257	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)})
65		intss1 4963	. . . . 5 ⊢ ({𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)} → ∩ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)} ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
66	64, 65	syl 17	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∩ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)} ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
67	1, 66	eqsstrd 4018	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (fi‘𝐹) ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
68	67	sselda 3983	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → 𝐴 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
69		sseq2 4010	. . . . 5 ⊢ (𝑡 = 𝐴 → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ 𝐴))
70	69	rexbidv 3179	. . . 4 ⊢ (𝑡 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))
71	70	elrab 3692	. . 3 ⊢ (𝐴 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ (𝐴 ∈ 𝒫 ∪ 𝐹 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))
72	71	simprbi 496	. 2 ⊢ (𝐴 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)
73	68, 72	syl 17	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ∩ cint 4946  ‘cfv 6561  ficfi 9450  fBascfbas 21352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-fbas 21361

This theorem is referenced by:  fbssint  23846  fbunfip  23877  fmfnfmlem1  23962  fmfnfmlem4  23965