Theorem tsmsfbas 24256

Description: The collection of all sets of the form 𝐹(𝑧) = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}, which can be read as the set of all finite subsets of 𝐴 which contain 𝑧 as a subset, for each finite subset 𝑧 of 𝐴, form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
tsmsfbas.s	⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsfbas.f	⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
tsmsfbas.l	⊢ 𝐿 = ran 𝐹
tsmsfbas.a	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
tsmsfbas	⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆))

Distinct variable groups:   𝑧,𝐴   𝑦,𝑧,𝑆

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑦)   𝐹(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem tsmsfbas

Dummy variables 𝑢 𝑎 𝑣 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsfbas.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
2		elex 3484	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		tsmsfbas.l	. . 3 ⊢ 𝐿 = ran 𝐹
4		ssrab2 4042	. . . . . . 7 ⊢ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆
5		tsmsfbas.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
6		pwexg 5352	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7		inex1g 5292	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
8	6, 7	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9	5, 8	eqeltrid 2873	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝑆 ∈ V)
10	9	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ V)
11		elpw2g 5306	. . . . . . . 8 ⊢ (𝑆 ∈ V → ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆))
12	10, 11	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆))
13	4, 12	mpbiri 261	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆)
14		tsmsfbas.f	. . . . . 6 ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
15	13, 14	fmptd 7112	. . . . 5 ⊢ (𝐴 ∈ V → 𝐹:𝑆⟶𝒫 𝑆)
16	15	frnd 6717	. . . 4 ⊢ (𝐴 ∈ V → ran 𝐹 ⊆ 𝒫 𝑆)
17		0ss 4364	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐴
18		0fi 9041	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
19		elfpw 9313	. . . . . . . . . 10 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin))
20	17, 18, 19	mpbir2an 723	. . . . . . . . 9 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
21	20, 5	eleqtrri 2868	. . . . . . . 8 ⊢ ∅ ∈ 𝑆
22		0ss 4364	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
23	22	rgenw 3089	. . . . . . . 8 ⊢ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦
24		rabid2 3456	. . . . . . . . . 10 ⊢ (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
25		sseq1 3970	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑧 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
26	25	ralbidv 3194	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦))
27	24, 26	bitrid 286	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦))
28	27	rspcev 3590	. . . . . . . 8 ⊢ ((∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦) → ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
29	21, 23, 28	mp2an 704	. . . . . . 7 ⊢ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}
30	14	elrnmpt 5951	. . . . . . . 8 ⊢ (𝑆 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
31	9, 30	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
32	29, 31	mpbiri 261	. . . . . 6 ⊢ (𝐴 ∈ V → 𝑆 ∈ ran 𝐹)
33	32	ne0d 4303	. . . . 5 ⊢ (𝐴 ∈ V → ran 𝐹 ≠ ∅)
34		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
35		ssid 3967	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ 𝑧
36		sseq2 3971	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑧))
37	36	rspcev 3590	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 ⊆ 𝑧) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
38	34, 35, 37	sylancl 597	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
39		rabn0 4353	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅ ↔ ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
40	38, 39	sylibr 237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅)
41	40	necomd 3019	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∅ ≠ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
42	41	neneqd 2969	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ¬ ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
43	42	nrexdv 3166	. . . . . . 7 ⊢ (𝐴 ∈ V → ¬ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
44		0ex 5274	. . . . . . . 8 ⊢ ∅ ∈ V
45	14	elrnmpt 5951	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
47	43, 46	sylnibr 332	. . . . . 6 ⊢ (𝐴 ∈ V → ¬ ∅ ∈ ran 𝐹)
48		df-nel 3071	. . . . . 6 ⊢ (∅ ∉ ran 𝐹 ↔ ¬ ∅ ∈ ran 𝐹)
49	47, 48	sylibr 237	. . . . 5 ⊢ (𝐴 ∈ V → ∅ ∉ ran 𝐹)
50		elfpw 9313	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑢 ⊆ 𝐴 ∧ 𝑢 ∈ Fin))
51	50	simplbi 501	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ⊆ 𝐴)
52	51, 5	eleq2s 2887	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ⊆ 𝐴)
53		elfpw 9313	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣 ⊆ 𝐴 ∧ 𝑣 ∈ Fin))
54	53	simplbi 501	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ⊆ 𝐴)
55	54, 5	eleq2s 2887	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ⊆ 𝐴)
56	52, 55	anim12i 624	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴))
57		unss 4151	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) ↔ (𝑢 ∪ 𝑣) ⊆ 𝐴)
58	56, 57	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ⊆ 𝐴)
59		elinel2 4163	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ∈ Fin)
60	59, 5	eleq2s 2887	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ Fin)
61		elinel2 4163	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ∈ Fin)
62	61, 5	eleq2s 2887	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ Fin)
63		unfi 9157	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Fin ∧ 𝑣 ∈ Fin) → (𝑢 ∪ 𝑣) ∈ Fin)
64	60, 62, 63	syl2an 607	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ Fin)
65		elfpw 9313	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑢 ∪ 𝑣) ⊆ 𝐴 ∧ (𝑢 ∪ 𝑣) ∈ Fin))
66	58, 64, 65	sylanbrc 594	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
67	66	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
68	67, 5	eleqtrrdi 2880	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ 𝑆)
69		eqidd 2770	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
70		sseq1 3970	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → (𝑎 ⊆ 𝑦 ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦))
71	70	rabbidv 3430	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
72	71	rspceeqv 3613	. . . . . . . . . . 11 ⊢ (((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) → ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
73	68, 69, 72	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
74	9	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → 𝑆 ∈ V)
75		rabexg 5310	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V)
76	74, 75	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V)
77		sseq1 3970	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑎 → (𝑧 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
78	77	rabbidv 3430	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑎 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
79	78	cbvmptv 5219	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
80	14, 79	eqtri 2792	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
81	80	elrnmpt 5951	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V → ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹 ↔ ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}))
82	76, 81	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹 ↔ ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}))
83	73, 82	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹)
84		pwidg 4587	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
85	76, 84	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
86		inelcm 4431	. . . . . . . . 9 ⊢ (({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹 ∧ {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) → (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
87	83, 85, 86	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
88	87	ralrimivva 3214	. . . . . . 7 ⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
89		rabexg 5310	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
90	9, 89	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
91	90	ralrimivw 3167	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
92		sseq1 3970	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑢 → (𝑧 ⊆ 𝑦 ↔ 𝑢 ⊆ 𝑦))
93	92	rabbidv 3430	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
94	93	cbvmptv 5219	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
95	14, 94	eqtri 2792	. . . . . . . . 9 ⊢ 𝐹 = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
96		ineq1 4174	. . . . . . . . . . . . . 14 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
97		inrab 4277	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)}
98		unss 4151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦) ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦)
99	98	rabbii 3428	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}
100	97, 99	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}
101	96, 100	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
102	101	pweqd 4584	. . . . . . . . . . . 12 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
103	102	ineq2d 4181	. . . . . . . . . . 11 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) = (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}))
104	103	neeq1d 3023	. . . . . . . . . 10 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅))
105	104	ralbidv 3194	. . . . . . . . 9 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅))
106	95, 105	ralrnmptw 7092	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V → (∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅))
107	91, 106	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅))
108	88, 107	mpbird 260	. . . . . 6 ⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)
109		rabexg 5310	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
110	9, 109	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
111	110	ralrimivw 3167	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∀𝑣 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
112		sseq1 3970	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (𝑧 ⊆ 𝑦 ↔ 𝑣 ⊆ 𝑦))
113	112	rabbidv 3430	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
114	113	cbvmptv 5219	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
115	14, 114	eqtri 2792	. . . . . . . . 9 ⊢ 𝐹 = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
116		ineq2 4175	. . . . . . . . . . . 12 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (𝑎 ∩ 𝑏) = (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
117	116	pweqd 4584	. . . . . . . . . . 11 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → 𝒫 (𝑎 ∩ 𝑏) = 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
118	117	ineq2d 4181	. . . . . . . . . 10 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) = (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})))
119	118	neeq1d 3023	. . . . . . . . 9 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
120	115, 119	ralrnmptw 7092	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
121	111, 120	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
122	121	ralbidv 3194	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
123	108, 122	mpbird 260	. . . . 5 ⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅)
124	33, 49, 123	3jca 1144	. . . 4 ⊢ (𝐴 ∈ V → (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅))
125		isfbas 23957	. . . . 5 ⊢ (𝑆 ∈ V → (ran 𝐹 ∈ (fBas‘𝑆) ↔ (ran 𝐹 ⊆ 𝒫 𝑆 ∧ (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅))))
126	9, 125	syl 18	. . . 4 ⊢ (𝐴 ∈ V → (ran 𝐹 ∈ (fBas‘𝑆) ↔ (ran 𝐹 ⊆ 𝒫 𝑆 ∧ (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅))))
127	16, 124, 126	mpbir2and 725	. . 3 ⊢ (𝐴 ∈ V → ran 𝐹 ∈ (fBas‘𝑆))
128	3, 127	eqeltrid 2873	. 2 ⊢ (𝐴 ∈ V → 𝐿 ∈ (fBas‘𝑆))
129	1, 2, 128	3syl 19	1 ⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∉ wnel 3070  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567   ↦ cmpt 5196  ran crn 5665  ‘cfv 6539  Fincfn 8945  fBascfbas 21481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7865  df-en 8946  df-fin 8949  df-fbas 21490

This theorem is referenced by:  eltsms  24261  haustsms  24264  tsmscls  24266  tsmsmhm  24274  tsmsadd  24275