Theorem fbssfi 21451

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 8212	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (fi‘𝐹) = ∩ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)})
2		inss1 3795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
3		simp1r 1079	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → 𝑢 ∈ 𝒫 ∪ 𝐹)
4	3	elpwid 4118	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → 𝑢 ⊆ ∪ 𝐹)
5	2, 4	syl5ss 3579	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐹)
6		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
7	6	inex1 4727	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝑣) ∈ V
8	7	elpw 4114	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∩ 𝑣) ∈ 𝒫 ∪ 𝐹 ↔ (𝑢 ∩ 𝑣) ⊆ ∪ 𝐹)
9	5, 8	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑢 ∩ 𝑣) ∈ 𝒫 ∪ 𝐹)
10		simpl 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
11		simpl 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) → 𝑦 ∈ 𝐹)
12		simpl 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣) → 𝑧 ∈ 𝐹)
13		fbasssin 21450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
14	10, 11, 12, 13	syl3an 1360	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
15		ss2in 3802	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ⊆ 𝑢 ∧ 𝑧 ⊆ 𝑣) → (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣))
16	15	ad2ant2l 778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣))
17	16	3adant1 1072	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣))
18		sstr 3576	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣)) → 𝑥 ⊆ (𝑢 ∩ 𝑣))
19	18	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ 𝑧) ⊆ (𝑢 ∩ 𝑣) → (𝑥 ⊆ (𝑦 ∩ 𝑧) → 𝑥 ⊆ (𝑢 ∩ 𝑣)))
20	17, 19	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → (𝑥 ⊆ (𝑦 ∩ 𝑧) → 𝑥 ⊆ (𝑢 ∩ 𝑣)))
21	20	reximdv 2999	. . . . . . . . . . . . . . . 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)))
22	14, 21	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑣)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑢 ∩ 𝑣))
23		sseq2 3590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝑢 ∩ 𝑣) → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ (𝑢 ∩ 𝑣)))
24	23	rexbidv 3034	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 ∩ 𝑣) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑢 ∩ 𝑣)))
25	24	elrab 3331	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ ((𝑢 ∩ 𝑣) ∈ 𝒫 ∪ 𝐹 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑢 ∩ 𝑣)))
26	9, 22, 25	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ (((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	3expa 1257	. . . . . . . . . . . . 13 ⊢ ((((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	rexlimdvaa 3014	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝒫 ∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑢)) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣 → (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
29	28	ralrimivw 2950	. . . . . . . . . . 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}))
30		sseq2 3590	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑣 → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ 𝑣))
31	30	rexbidv 3034	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑣 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑣))
32		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝑣 ↔ 𝑧 ⊆ 𝑣))
33	32	cbvrexv 3148	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑣 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣)
34	31, 33	syl6bb 275	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑣))
35	34	ralrab 3335	. . . . . . . . . . 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}))
36	29, 35	sylibr 223	. . . . . . . . . 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})
37	36	rexlimdvaa 3014	. . . . . . . . 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}))
38	37	ralrimiva 2949	. . . . . . . 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}))
39		sseq2 3590	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑢 → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ 𝑢))
40	39	rexbidv 3034	. . . . . . . . . 10 ⊢ (𝑡 = 𝑢 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑢))
41		sseq1 3589	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑢 ↔ 𝑦 ⊆ 𝑢))
42	41	cbvrexv 3148	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑢 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢)
43	40, 42	syl6bb 275	. . . . . . . . 9 ⊢ (𝑡 = 𝑢 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑢))
44	43	ralrab 3335	. . . . . . . 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}))
45	38, 44	sylibr 223	. . . . . . 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})
46		pwuni 4825	. . . . . . . 8 ⊢ 𝐹 ⊆ 𝒫 ∪ 𝐹
47		ssid 3587	. . . . . . . . . 10 ⊢ 𝑡 ⊆ 𝑡
48		sseq1 3589	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡))
49	48	rspcev 3282	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)
50	47, 49	mpan2 703	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)
51	50	rgen 2906	. . . . . . . 8 ⊢ ∀𝑡 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡
52		ssrab 3643	. . . . . . . 8 ⊢ (𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ (𝐹 ⊆ 𝒫 ∪ 𝐹 ∧ ∀𝑡 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
53	46, 51, 52	mpbir2an 957	. . . . . . 7 ⊢ 𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}
54	45, 53	jctil 558	. . . . . 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}))
55		uniexg 6853	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∪ 𝐹 ∈ V)
56		pwexg 4776	. . . . . . 7 ⊢ (∪ 𝐹 ∈ V → 𝒫 ∪ 𝐹 ∈ V)
57		rabexg 4739	. . . . . . 7 ⊢ (𝒫 ∪ 𝐹 ∈ V → {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ V)
58		sseq2 3590	. . . . . . . . 9 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → (𝐹 ⊆ 𝑧 ↔ 𝐹 ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
59		eleq2 2677	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → ((𝑢 ∩ 𝑣) ∈ 𝑧 ↔ (𝑢 ∩ 𝑣) ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡}))
60	59	raleqbi1dv 3123	. . . . . . . . . 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}))
61	60	raleqbi1dv 3123	. . . . . . . . 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}))
62	58, 61	anbi12d 743	. . . . . . . 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})))
63	62	elabg 3320	. . . . . . 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})))
64	55, 56, 57, 63	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})))
65	54, 64	mpbird 246	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ∈ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)})
66		intss1 4427	. . . . 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})
67	65, 66	syl 17	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∩ {𝑧 ∣ (𝐹 ⊆ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (𝑢 ∩ 𝑣) ∈ 𝑧)} ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
68	1, 67	eqsstrd 3602	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (fi‘𝐹) ⊆ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
69	68	sselda 3568	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → 𝐴 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡})
70		sseq2 3590	. . . . 5 ⊢ (𝑡 = 𝐴 → (𝑥 ⊆ 𝑡 ↔ 𝑥 ⊆ 𝐴))
71	70	rexbidv 3034	. . . 4 ⊢ (𝑡 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))
72	71	elrab 3331	. . 3 ⊢ (𝐴 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} ↔ (𝐴 ∈ 𝒫 ∪ 𝐹 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴))
73	72	simprbi 479	. 2 ⊢ (𝐴 ∈ {𝑡 ∈ 𝒫 ∪ 𝐹 ∣ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡} → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)
74	69, 73	syl 17	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ‘cfv 5804  ficfi 8199  fBascfbas 19555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-fbas 19564

This theorem is referenced by:  fbssint  21452  fbunfip  21483  fmfnfmlem1  21568  fmfnfmlem4  21571