Theorem tsmsfbas 24152

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 3499	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		tsmsfbas.l	. . 3 ⊢ 𝐿 = ran 𝐹
4		ssrab2 4090	. . . . . . 7 ⊢ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆
5		tsmsfbas.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
6		pwexg 5384	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7		inex1g 5325	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9	5, 8	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝑆 ∈ V)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ V)
11		elpw2g 5339	. . . . . . . 8 ⊢ (𝑆 ∈ V → ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆))
12	10, 11	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆))
13	4, 12	mpbiri 258	. . . . . 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 7134	. . . . 5 ⊢ (𝐴 ∈ V → 𝐹:𝑆⟶𝒫 𝑆)
16	15	frnd 6745	. . . 4 ⊢ (𝐴 ∈ V → ran 𝐹 ⊆ 𝒫 𝑆)
17		0ss 4406	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐴
18		0fi 9081	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
19		elfpw 9392	. . . . . . . . . 10 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin))
20	17, 18, 19	mpbir2an 711	. . . . . . . . 9 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
21	20, 5	eleqtrri 2838	. . . . . . . 8 ⊢ ∅ ∈ 𝑆
22		0ss 4406	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
23	22	rgenw 3063	. . . . . . . 8 ⊢ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦
24		rabid2 3468	. . . . . . . . . 10 ⊢ (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
25		sseq1 4021	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑧 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
26	25	ralbidv 3176	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦))
27	24, 26	bitrid 283	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦))
28	27	rspcev 3622	. . . . . . . 8 ⊢ ((∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦) → ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
29	21, 23, 28	mp2an 692	. . . . . . 7 ⊢ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}
30	14	elrnmpt 5972	. . . . . . . 8 ⊢ (𝑆 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
31	9, 30	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
32	29, 31	mpbiri 258	. . . . . 6 ⊢ (𝐴 ∈ V → 𝑆 ∈ ran 𝐹)
33	32	ne0d 4348	. . . . 5 ⊢ (𝐴 ∈ V → ran 𝐹 ≠ ∅)
34		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
35		ssid 4018	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ 𝑧
36		sseq2 4022	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑧))
37	36	rspcev 3622	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 ⊆ 𝑧) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
38	34, 35, 37	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
39		rabn0 4395	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅ ↔ ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
40	38, 39	sylibr 234	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅)
41	40	necomd 2994	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∅ ≠ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
42	41	neneqd 2943	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ¬ ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
43	42	nrexdv 3147	. . . . . . 7 ⊢ (𝐴 ∈ V → ¬ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
44		0ex 5313	. . . . . . . 8 ⊢ ∅ ∈ V
45	14	elrnmpt 5972	. . . . . . . 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 329	. . . . . 6 ⊢ (𝐴 ∈ V → ¬ ∅ ∈ ran 𝐹)
48		df-nel 3045	. . . . . 6 ⊢ (∅ ∉ ran 𝐹 ↔ ¬ ∅ ∈ ran 𝐹)
49	47, 48	sylibr 234	. . . . 5 ⊢ (𝐴 ∈ V → ∅ ∉ ran 𝐹)
50		elfpw 9392	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑢 ⊆ 𝐴 ∧ 𝑢 ∈ Fin))
51	50	simplbi 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ⊆ 𝐴)
52	51, 5	eleq2s 2857	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ⊆ 𝐴)
53		elfpw 9392	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣 ⊆ 𝐴 ∧ 𝑣 ∈ Fin))
54	53	simplbi 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ⊆ 𝐴)
55	54, 5	eleq2s 2857	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ⊆ 𝐴)
56	52, 55	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴))
57		unss 4200	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) ↔ (𝑢 ∪ 𝑣) ⊆ 𝐴)
58	56, 57	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ⊆ 𝐴)
59		elinel2 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ∈ Fin)
60	59, 5	eleq2s 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ Fin)
61		elinel2 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ∈ Fin)
62	61, 5	eleq2s 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ Fin)
63		unfi 9210	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Fin ∧ 𝑣 ∈ Fin) → (𝑢 ∪ 𝑣) ∈ Fin)
64	60, 62, 63	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ Fin)
65		elfpw 9392	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑢 ∪ 𝑣) ⊆ 𝐴 ∧ (𝑢 ∪ 𝑣) ∈ Fin))
66	58, 64, 65	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
67	66	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
68	67, 5	eleqtrrdi 2850	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ 𝑆)
69		eqidd 2736	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
70		sseq1 4021	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → (𝑎 ⊆ 𝑦 ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦))
71	70	rabbidv 3441	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
72	71	rspceeqv 3645	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
74	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → 𝑆 ∈ V)
75		rabexg 5343	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V)
77		sseq1 4021	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑎 → (𝑧 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
78	77	rabbidv 3441	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑎 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
79	78	cbvmptv 5261	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
80	14, 79	eqtri 2763	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
81	80	elrnmpt 5972	. . . . . . . . . . 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 17	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹)
84		pwidg 4625	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
85	76, 84	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
86		inelcm 4471	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
88	87	ralrimivva 3200	. . . . . . 7 ⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
89		rabexg 5343	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
90	9, 89	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
91	90	ralrimivw 3148	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
92		sseq1 4021	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑢 → (𝑧 ⊆ 𝑦 ↔ 𝑢 ⊆ 𝑦))
93	92	rabbidv 3441	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
94	93	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
95	14, 94	eqtri 2763	. . . . . . . . 9 ⊢ 𝐹 = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
96		ineq1 4221	. . . . . . . . . . . . . 14 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
97		inrab 4322	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)}
98		unss 4200	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦) ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦)
99	98	rabbii 3439	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}
100	97, 99	eqtri 2763	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}
101	96, 100	eqtrdi 2791	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
102	101	pweqd 4622	. . . . . . . . . . . 12 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
103	102	ineq2d 4228	. . . . . . . . . . 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 2998	. . . . . . . . . 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 3176	. . . . . . . . 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 7114	. . . . . . . 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 17	. . . . . . 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 257	. . . . . 6 ⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)
109		rabexg 5343	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
110	9, 109	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
111	110	ralrimivw 3148	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∀𝑣 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
112		sseq1 4021	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (𝑧 ⊆ 𝑦 ↔ 𝑣 ⊆ 𝑦))
113	112	rabbidv 3441	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
114	113	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
115	14, 114	eqtri 2763	. . . . . . . . 9 ⊢ 𝐹 = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
116		ineq2 4222	. . . . . . . . . . . 12 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (𝑎 ∩ 𝑏) = (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
117	116	pweqd 4622	. . . . . . . . . . 11 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → 𝒫 (𝑎 ∩ 𝑏) = 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
118	117	ineq2d 4228	. . . . . . . . . 10 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) = (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})))
119	118	neeq1d 2998	. . . . . . . . 9 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
120	115, 119	ralrnmptw 7114	. . . . . . . 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 17	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
122	121	ralbidv 3176	. . . . . 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 257	. . . . 5 ⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅)
124	33, 49, 123	3jca 1127	. . . 4 ⊢ (𝐴 ∈ V → (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅))
125		isfbas 23853	. . . . 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 17	. . . 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 713	. . 3 ⊢ (𝐴 ∈ V → ran 𝐹 ∈ (fBas‘𝑆))
128	3, 127	eqeltrid 2843	. 2 ⊢ (𝐴 ∈ V → 𝐿 ∈ (fBas‘𝑆))
129	1, 2, 128	3syl 18	1 ⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∉ wnel 3044  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605   ↦ cmpt 5231  ran crn 5690  ‘cfv 6563  Fincfn 8984  fBascfbas 21370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-en 8985  df-fin 8988  df-fbas 21379

This theorem is referenced by:  eltsms  24157  haustsms  24160  tsmscls  24162  tsmsmhm  24170  tsmsadd  24171