Theorem fbun 22899

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 4106	. . . . 5 ⊢ (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝐹 ∪ 𝐺))
2		elun2 4107	. . . . 5 ⊢ (𝑦 ∈ 𝐺 → 𝑦 ∈ (𝐹 ∪ 𝐺))
3	1, 2	anim12i 612	. . . 4 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐺) → (𝑥 ∈ (𝐹 ∪ 𝐺) ∧ 𝑦 ∈ (𝐹 ∪ 𝐺)))
4		fbasssin 22895	. . . . 5 ⊢ (((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (𝐹 ∪ 𝐺) ∧ 𝑦 ∈ (𝐹 ∪ 𝐺)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
5	4	3expb 1118	. . . 4 ⊢ (((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ (𝐹 ∪ 𝐺) ∧ 𝑦 ∈ (𝐹 ∪ 𝐺))) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
6	3, 5	sylan2 592	. . 3 ⊢ (((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐺)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
7	6	ralrimivva 3114	. 2 ⊢ ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
8		fbsspw 22891	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
9	8	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
10		fbsspw 22891	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → 𝐺 ⊆ 𝒫 𝑋)
11	10	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐺 ⊆ 𝒫 𝑋)
12	9, 11	unssd 4116	. . . . 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 4102	. . . . . . . 8 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
15		fbasne0 22889	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
16		ssn0 4331	. . . . . . . 8 ⊢ ((𝐹 ⊆ (𝐹 ∪ 𝐺) ∧ 𝐹 ≠ ∅) → (𝐹 ∪ 𝐺) ≠ ∅)
17	14, 15, 16	sylancr 586	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∪ 𝐺) ≠ ∅)
18	17	adantr 480	. . . . . 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 22890	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21		0nelfb 22890	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐺)
22		df-nel 3049	. . . . . . . . 9 ⊢ (∅ ∉ (𝐹 ∪ 𝐺) ↔ ¬ ∅ ∈ (𝐹 ∪ 𝐺))
23		elun 4079	. . . . . . . . . 10 ⊢ (∅ ∈ (𝐹 ∪ 𝐺) ↔ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
24	23	notbii 319	. . . . . . . . 9 ⊢ (¬ ∅ ∈ (𝐹 ∪ 𝐺) ↔ ¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
25		ioran 980	. . . . . . . . 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 595	. . . . . 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 22895	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
31		ssrexv 3984	. . . . . . . . . . . . 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 1118	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
34	33	ralrimivva 3114	. . . . . . . . . 10 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
35	34	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
36		pm3.2 469	. . . . . . . . 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 3094	. . . . . . . . 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 4122	. . . . . . . . . 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 3280	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑦 ∈ 𝐺 ∀𝑥 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
44		ineq1 4136	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑥 ∩ 𝑦) = (𝑤 ∩ 𝑦))
45	44	sseq2d 3949	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ 𝑧 ⊆ (𝑤 ∩ 𝑦)))
46	45	rexbidv 3225	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦)))
47	46	cbvralvw 3372	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑤 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦))
48	47	ralbii 3090	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐺 ∀𝑥 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ ∀𝑦 ∈ 𝐺 ∀𝑤 ∈ 𝐹 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦))
49		ineq2 4137	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑤 ∩ 𝑦) = (𝑤 ∩ 𝑥))
50	49	sseq2d 3949	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑧 ⊆ (𝑤 ∩ 𝑦) ↔ 𝑧 ⊆ (𝑤 ∩ 𝑥)))
51	50	rexbidv 3225	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑦) ↔ ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑥)))
52		ineq1 4136	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (𝑤 ∩ 𝑥) = (𝑦 ∩ 𝑥))
53		incom 4131	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
54	52, 53	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → (𝑤 ∩ 𝑥) = (𝑥 ∩ 𝑦))
55	54	sseq2d 3949	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑧 ⊆ (𝑤 ∩ 𝑥) ↔ 𝑧 ⊆ (𝑥 ∩ 𝑦)))
56	55	rexbidv 3225	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑤 ∩ 𝑥) ↔ ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
57	51, 56	cbvral2vw 3385	. . . . . . . . . . . 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 4103	. . . . . . . . . . . . . 14 ⊢ 𝐺 ⊆ (𝐹 ∪ 𝐺)
61		fbasssin 22895	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → ∃𝑧 ∈ 𝐺 𝑧 ⊆ (𝑥 ∩ 𝑦))
62		ssrexv 3984	. . . . . . . . . . . . . 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 1118	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ (fBas‘𝑋) ∧ (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺)) → ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
65	64	ralrimivva 3114	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (fBas‘𝑋) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
66	65	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥 ∈ 𝐺 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))
67	59, 66	anim12i 612	. . . . . . . . 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 413	. . . . . . . 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 3094	. . . . . . . . 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 512	. . . . . 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 4122	. . . . . 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 1127	. . . 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 512	. . 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 6788	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
79	78	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝑋 ∈ dom fBas)
80		isfbas2 22894	. . . 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 395   ∨ wo 843   ∧ w3a 1085   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048  ∀wral 3063  ∃wrex 3064   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  dom cdm 5580  ‘cfv 6418  fBascfbas 20498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-fbas 20507

This theorem is referenced by: (None)