Theorem fbun 23191

Description: A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fbun	⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐹,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem fbun

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun1 4136	. . . . 5 ⊢ (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝐹 ∪ 𝐺))
2		elun2 4137	. . . . 5 ⊢ (𝑦 ∈ 𝐺 → 𝑦 ∈ (𝐹 ∪ 𝐺))
3	1, 2	anim12i 613	. . . 4 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐺) → (𝑥 ∈ (𝐹 ∪ 𝐺) ∧ 𝑦 ∈ (𝐹 ∪ 𝐺)))
4		fbasssin 23187	. . . . 5 ⊢ (((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (𝐹 ∪ 𝐺) ∧ 𝑦 ∈ (𝐹 ∪ 𝐺)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
5	4	3expb 1120	. . . 4 ⊢ (((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ (𝐹 ∪ 𝐺) ∧ 𝑦 ∈ (𝐹 ∪ 𝐺))) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
6	3, 5	sylan2 593	. . 3 ⊢ (((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐺)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
7	6	ralrimivva 3197	. 2 ⊢ ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
8		fbsspw 23183	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
9	8	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
10		fbsspw 23183	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → 𝐺 ⊆ 𝒫 𝑋)
11	10	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐺 ⊆ 𝒫 𝑋)
12	9, 11	unssd 4146	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝐹 ∪ 𝐺) ⊆ 𝒫 𝑋)
13	12	a1d 25	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (𝐹 ∪ 𝐺) ⊆ 𝒫 𝑋))
14		ssun1 4132	. . . . . . . 8 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
15		fbasne0 23181	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
16		ssn0 4360	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐹 ∪ 𝐺) ∧ 𝐹 ≠ ∅) → (𝐹 ∪ 𝐺) ≠ ∅)
17	14, 15, 16	sylancr 587	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∪ 𝐺) ≠ ∅)
18	17	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝐹 ∪ 𝐺) ≠ ∅)
19	18	a1d 25	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (𝐹 ∪ 𝐺) ≠ ∅))
20		0nelfb 23182	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21		0nelfb 23182	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐺)
22		df-nel 3050	. . . . . . . . 9 ⊢ (∅ ∉ (𝐹 ∪ 𝐺) ↔ ¬ ∅ ∈ (𝐹 ∪ 𝐺))
23		elun 4108	. . . . . . . . . 10 ⊢ (∅ ∈ (𝐹 ∪ 𝐺) ↔ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
24	23	notbii 319	. . . . . . . . 9 ⊢ (¬ ∅ ∈ (𝐹 ∪ 𝐺) ↔ ¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
25		ioran 982	. . . . . . . . 9 ⊢ (¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
26	22, 24, 25	3bitri 296	. . . . . . . 8 ⊢ (∅ ∉ (𝐹 ∪ 𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
27	26	biimpri 227	. . . . . . 7 ⊢ ((¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺) → ∅ ∉ (𝐹 ∪ 𝐺))
28	20, 21, 27	syl2an 596	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∅ ∉ (𝐹 ∪ 𝐺))
29	28	a1d 25	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ∅ ∉ (𝐹 ∪ 𝐺)))
30		fbasssin 23187	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
31		ssrexv 4011	. . . . . . . . . . . . 13 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
32	14, 30, 31	mpsyl 68	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
33	32	3expb 1120	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
34	33	ralrimivva 3197	. . . . . . . . . 10 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
35	34	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
36		pm3.2 470	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))))
37	35, 36	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))))
38		r19.26 3114	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐹 (∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) ↔ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
39		ralun 4152	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) → ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
40	39	ralimi 3086	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐹 (∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
41	38, 40	sylbir 234	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
42	37, 41	syl6 35	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
43		ralcom 3272	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑦 ∈ 𝐺 ∀𝑥 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
44		ineq1 4165	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑥 ∩ 𝑦) = (𝑤 ∩ 𝑦))
45	44	sseq2d 3976	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ 𝑧 ⊆ (𝑤 ∩ 𝑦)))
46	45	rexbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦)))
47	46	cbvralvw 3225	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑤 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦))
48	47	ralbii 3096	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐺 ∀𝑥 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑦 ∈ 𝐺 ∀𝑤 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦))
49		ineq2 4166	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑤 ∩ 𝑦) = (𝑤 ∩ 𝑥))
50	49	sseq2d 3976	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑧 ⊆ (𝑤 ∩ 𝑦) ↔ 𝑧 ⊆ (𝑤 ∩ 𝑥)))
51	50	rexbidv 3175	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦) ↔ ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑥)))
52		ineq1 4165	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (𝑤 ∩ 𝑥) = (𝑦 ∩ 𝑥))
53		incom 4161	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
54	52, 53	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → (𝑤 ∩ 𝑥) = (𝑥 ∩ 𝑦))
55	54	sseq2d 3976	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑧 ⊆ (𝑤 ∩ 𝑥) ↔ 𝑧 ⊆ (𝑥 ∩ 𝑦)))
56	55	rexbidv 3175	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑥) ↔ ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
57	51, 56	cbvral2vw 3227	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐺 ∀𝑤 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦) ↔ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
58	43, 48, 57	3bitri 296	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
59	58	biimpi 215	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
60		ssun2 4133	. . . . . . . . . . . . . 14 ⊢ 𝐺 ⊆ (𝐹 ∪ 𝐺)
61		fbasssin 23187	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → ∃𝑧 ∈ 𝐺 𝑧 ⊆ (𝑥 ∩ 𝑦))
62		ssrexv 4011	. . . . . . . . . . . . . 14 ⊢ (𝐺 ⊆ (𝐹 ∪ 𝐺) → (∃𝑧 ∈ 𝐺 𝑧 ⊆ (𝑥 ∩ 𝑦) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
63	60, 61, 62	mpsyl 68	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
64	63	3expb 1120	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
65	64	ralrimivva 3197	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (fBas‘𝑋) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
66	65	adantl 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
67	59, 66	anim12i 613	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ (𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋))) → (∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
68	67	expcom 414	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))))
69		r19.26 3114	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐺 (∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) ↔ (∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
70	39	ralimi 3086	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐺 (∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
71	69, 70	sylbir 234	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
72	68, 71	syl6 35	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
73	42, 72	jcad 513	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))))
74		ralun 4152	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ∧ ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)) → ∀𝑥 ∈ (𝐹 ∪ 𝐺)∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
75	73, 74	syl6 35	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ∀𝑥 ∈ (𝐹 ∪ 𝐺)∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
76	19, 29, 75	3jcad 1129	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ((𝐹 ∪ 𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹 ∪ 𝐺) ∧ ∀𝑥 ∈ (𝐹 ∪ 𝐺)∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))))
77	13, 76	jcad 513	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → ((𝐹 ∪ 𝐺) ⊆ 𝒫 𝑋 ∧ ((𝐹 ∪ 𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹 ∪ 𝐺) ∧ ∀𝑥 ∈ (𝐹 ∪ 𝐺)∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))))
78		elfvdm 6879	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
79	78	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝑋 ∈ dom fBas)
80		isfbas2 23186	. . . 4 ⊢ (𝑋 ∈ dom fBas → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ 𝐺) ⊆ 𝒫 𝑋 ∧ ((𝐹 ∪ 𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹 ∪ 𝐺) ∧ ∀𝑥 ∈ (𝐹 ∪ 𝐺)∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))))
81	79, 80	syl 17	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ 𝐺) ⊆ 𝒫 𝑋 ∧ ((𝐹 ∪ 𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹 ∪ 𝐺) ∧ ∀𝑥 ∈ (𝐹 ∪ 𝐺)∀𝑦 ∈ (𝐹 ∪ 𝐺)∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))))
82	77, 81	sylibrd 258	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) → (𝐹 ∪ 𝐺) ∈ (fBas‘𝑋)))
83	7, 82	impbid2 225	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  ∃wrex 3073   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  dom cdm 5633  ‘cfv 6496  fBascfbas 20784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-fbas 20793

This theorem is referenced by: (None)