Theorem tsmsfbas 21931

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 3212	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		tsmsfbas.l	. . 3 ⊢ 𝐿 = ran 𝐹
4		ssrab2 3687	. . . . . . 7 ⊢ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆
5		tsmsfbas.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
6		pwexg 4850	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7		inex1g 4801	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9	5, 8	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝑆 ∈ V)
10	9	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ V)
11		elpw2g 4827	. . . . . . . 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 248	. . . . . 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 6385	. . . . 5 ⊢ (𝐴 ∈ V → 𝐹:𝑆⟶𝒫 𝑆)
16		frn 6053	. . . . 5 ⊢ (𝐹:𝑆⟶𝒫 𝑆 → ran 𝐹 ⊆ 𝒫 𝑆)
17	15, 16	syl 17	. . . 4 ⊢ (𝐴 ∈ V → ran 𝐹 ⊆ 𝒫 𝑆)
18		0ss 3972	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐴
19		0fin 8188	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
20		elfpw 8268	. . . . . . . . . 10 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin))
21	18, 19, 20	mpbir2an 955	. . . . . . . . 9 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
22	21, 5	eleqtrri 2700	. . . . . . . 8 ⊢ ∅ ∈ 𝑆
23		0ss 3972	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
24	23	rgenw 2924	. . . . . . . 8 ⊢ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦
25		rabid2 3118	. . . . . . . . . 10 ⊢ (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
26		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑧 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
27	26	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦))
28	25, 27	syl5bb 272	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦))
29	28	rspcev 3309	. . . . . . . 8 ⊢ ((∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦) → ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
30	22, 24, 29	mp2an 708	. . . . . . 7 ⊢ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}
31	14	elrnmpt 5372	. . . . . . . 8 ⊢ (𝑆 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
32	9, 31	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
33	30, 32	mpbiri 248	. . . . . 6 ⊢ (𝐴 ∈ V → 𝑆 ∈ ran 𝐹)
34		ne0i 3921	. . . . . 6 ⊢ (𝑆 ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
35	33, 34	syl 17	. . . . 5 ⊢ (𝐴 ∈ V → ran 𝐹 ≠ ∅)
36		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
37		ssid 3624	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ 𝑧
38		sseq2 3627	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑧))
39	38	rspcev 3309	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 ⊆ 𝑧) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
40	36, 37, 39	sylancl 694	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
41		rabn0 3958	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅ ↔ ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)
42	40, 41	sylibr 224	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅)
43	42	necomd 2849	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∅ ≠ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
44	43	neneqd 2799	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ¬ ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
45	44	nrexdv 3001	. . . . . . 7 ⊢ (𝐴 ∈ V → ¬ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
46		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
47	14	elrnmpt 5372	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
48	46, 47	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
49	45, 48	sylnibr 319	. . . . . 6 ⊢ (𝐴 ∈ V → ¬ ∅ ∈ ran 𝐹)
50		df-nel 2898	. . . . . 6 ⊢ (∅ ∉ ran 𝐹 ↔ ¬ ∅ ∈ ran 𝐹)
51	49, 50	sylibr 224	. . . . 5 ⊢ (𝐴 ∈ V → ∅ ∉ ran 𝐹)
52		elfpw 8268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑢 ⊆ 𝐴 ∧ 𝑢 ∈ Fin))
53	52	simplbi 476	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ⊆ 𝐴)
54	53, 5	eleq2s 2719	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ⊆ 𝐴)
55		elfpw 8268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣 ⊆ 𝐴 ∧ 𝑣 ∈ Fin))
56	55	simplbi 476	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ⊆ 𝐴)
57	56, 5	eleq2s 2719	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ⊆ 𝐴)
58	54, 57	anim12i 590	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴))
59		unss 3787	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) ↔ (𝑢 ∪ 𝑣) ⊆ 𝐴)
60	58, 59	sylib 208	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ⊆ 𝐴)
61	52	simprbi 480	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ∈ Fin)
62	61, 5	eleq2s 2719	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ Fin)
63	55	simprbi 480	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ∈ Fin)
64	63, 5	eleq2s 2719	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ Fin)
65		unfi 8227	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Fin ∧ 𝑣 ∈ Fin) → (𝑢 ∪ 𝑣) ∈ Fin)
66	62, 64, 65	syl2an 494	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ Fin)
67		elfpw 8268	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑢 ∪ 𝑣) ⊆ 𝐴 ∧ (𝑢 ∪ 𝑣) ∈ Fin))
68	60, 66, 67	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
69	68	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
70	69, 5	syl6eleqr 2712	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ 𝑆)
71		eqidd 2623	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
72		sseq1 3626	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → (𝑎 ⊆ 𝑦 ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦))
73	72	rabbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
74	73	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑢 ∪ 𝑣) → ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦} ↔ {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}))
75	74	rspcev 3309	. . . . . . . . . . 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})
76	70, 71, 75	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
77	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → 𝑆 ∈ V)
78		rabexg 4812	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V)
79	77, 78	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V)
80		sseq1 3626	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑎 → (𝑧 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
81	80	rabbidv 3189	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑎 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
82	81	cbvmptv 4750	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
83	14, 82	eqtri 2644	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})
84	83	elrnmpt 5372	. . . . . . . . . . 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}))
85	79, 84	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}))
86	76, 85	mpbird 247	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹)
87		pwidg 4173	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
88	79, 87	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
89		inelcm 4032	. . . . . . . . 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}) ≠ ∅)
90	86, 88, 89	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
91	90	ralrimivva 2971	. . . . . . 7 ⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)
92		rabexg 4812	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
93	9, 92	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
94	93	ralrimivw 2967	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V)
95		sseq1 3626	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑢 → (𝑧 ⊆ 𝑦 ↔ 𝑢 ⊆ 𝑦))
96	95	rabbidv 3189	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
97	96	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
98	14, 97	eqtri 2644	. . . . . . . . 9 ⊢ 𝐹 = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦})
99		ineq1 3807	. . . . . . . . . . . . . 14 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
100		inrab 3899	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)}
101		unss 3787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦) ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦)
102	101	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑆 → ((𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦) ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦))
103	102	rabbiia 3185	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}
104	100, 103	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}
105	99, 104	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
106	105	pweqd 4163	. . . . . . . . . . . 12 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})
107	106	ineq2d 3814	. . . . . . . . . . 11 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) = (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}))
108	107	neeq1d 2853	. . . . . . . . . 10 ⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅))
109	108	ralbidv 2986	. . . . . . . . 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}) ≠ ∅))
110	98, 109	ralrnmpt 6368	. . . . . . . 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}) ≠ ∅))
111	94, 110	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}) ≠ ∅))
112	91, 111	mpbird 247	. . . . . 6 ⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)
113		rabexg 4812	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
114	9, 113	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
115	114	ralrimivw 2967	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∀𝑣 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V)
116		sseq1 3626	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (𝑧 ⊆ 𝑦 ↔ 𝑣 ⊆ 𝑦))
117	116	rabbidv 3189	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
118	117	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
119	14, 118	eqtri 2644	. . . . . . . . 9 ⊢ 𝐹 = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})
120		ineq2 3808	. . . . . . . . . . . 12 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (𝑎 ∩ 𝑏) = (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
121	120	pweqd 4163	. . . . . . . . . . 11 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → 𝒫 (𝑎 ∩ 𝑏) = 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))
122	121	ineq2d 3814	. . . . . . . . . 10 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) = (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})))
123	122	neeq1d 2853	. . . . . . . . 9 ⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
124	119, 123	ralrnmpt 6368	. . . . . . . 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})) ≠ ∅))
125	115, 124	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅))
126	125	ralbidv 2986	. . . . . 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})) ≠ ∅))
127	112, 126	mpbird 247	. . . . 5 ⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅)
128	35, 51, 127	3jca 1242	. . . 4 ⊢ (𝐴 ∈ V → (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅))
129		isfbas 21633	. . . . 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)) ≠ ∅))))
130	9, 129	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)) ≠ ∅))))
131	17, 128, 130	mpbir2and 957	. . 3 ⊢ (𝐴 ∈ V → ran 𝐹 ∈ (fBas‘𝑆))
132	3, 131	syl5eqel 2705	. 2 ⊢ (𝐴 ∈ V → 𝐿 ∈ (fBas‘𝑆))
133	1, 2, 132	3syl 18	1 ⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888  Fincfn 7955  fBascfbas 19734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fbas 19743

This theorem is referenced by:  eltsms  21936  haustsms  21939  tsmscls  21941  tsmsmhm  21949  tsmsadd  21950