Theorem txbas 21879

Description: The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
txval.1	⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))

Assertion

Ref	Expression
txbas	⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem txbas

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑝 𝑡 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txval.1	. . . . . . . 8 ⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
2		xpeq1 5421	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑦) = (𝑎 × 𝑦))
3		xpeq2 5428	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑎 × 𝑦) = (𝑎 × 𝑏))
4	2, 3	cbvmpov 7065	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑎 ∈ 𝑅, 𝑏 ∈ 𝑆 ↦ (𝑎 × 𝑏))
5	4	rnmpo 7100	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑢 ∣ ∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏)}
6	1, 5	eqtri 2802	. . . . . . 7 ⊢ 𝐵 = {𝑢 ∣ ∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏)}
7	6	abeq2i 2900	. . . . . 6 ⊢ (𝑢 ∈ 𝐵 ↔ ∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏))
8		xpeq1 5421	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → (𝑥 × 𝑦) = (𝑐 × 𝑦))
9		xpeq2 5428	. . . . . . . . . 10 ⊢ (𝑦 = 𝑑 → (𝑐 × 𝑦) = (𝑐 × 𝑑))
10	8, 9	cbvmpov 7065	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑐 ∈ 𝑅, 𝑑 ∈ 𝑆 ↦ (𝑐 × 𝑑))
11	10	rnmpo 7100	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑣 ∣ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)}
12	1, 11	eqtri 2802	. . . . . . 7 ⊢ 𝐵 = {𝑣 ∣ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)}
13	12	abeq2i 2900	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↔ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑))
14	7, 13	anbi12i 617	. . . . 5 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ↔ (∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
15		reeanv 3308	. . . . 5 ⊢ (∃𝑎 ∈ 𝑅 ∃𝑐 ∈ 𝑅 (∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)) ↔ (∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
16	14, 15	bitr4i 270	. . . 4 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ↔ ∃𝑎 ∈ 𝑅 ∃𝑐 ∈ 𝑅 (∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
17		reeanv 3308	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑆 ∃𝑑 ∈ 𝑆 (𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) ↔ (∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
18		basis2 21263	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ TopBases ∧ 𝑎 ∈ 𝑅) ∧ (𝑐 ∈ 𝑅 ∧ 𝑢 ∈ (𝑎 ∩ 𝑐))) → ∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)))
19	18	exp43 429	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ TopBases → (𝑎 ∈ 𝑅 → (𝑐 ∈ 𝑅 → (𝑢 ∈ (𝑎 ∩ 𝑐) → ∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐))))))
20	19	imp42 419	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ TopBases ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ 𝑢 ∈ (𝑎 ∩ 𝑐)) → ∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)))
21		basis2 21263	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ TopBases ∧ 𝑏 ∈ 𝑆) ∧ (𝑑 ∈ 𝑆 ∧ 𝑣 ∈ (𝑏 ∩ 𝑑))) → ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑)))
22	21	exp43 429	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ TopBases → (𝑏 ∈ 𝑆 → (𝑑 ∈ 𝑆 → (𝑣 ∈ (𝑏 ∩ 𝑑) → ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))))))
23	22	imp42 419	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ TopBases ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ 𝑣 ∈ (𝑏 ∩ 𝑑)) → ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑)))
24		reeanv 3308	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ((𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) ↔ (∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))))
25		opelxpi 5444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) → ⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦))
26		xpss12 5422	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ (𝑎 ∩ 𝑐) ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑)) → (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))
27	25, 26	anim12i 603	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) ∧ (𝑥 ⊆ (𝑎 ∩ 𝑐) ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
28	27	an4s 647	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
29	28	reximi 3190	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦 ∈ 𝑆 ((𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
30	29	reximi 3190	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ((𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
31	24, 30	sylbir 227	. . . . . . . . . . . . . 14 ⊢ ((∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
32	20, 23, 31	syl2an 586	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ TopBases ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ 𝑢 ∈ (𝑎 ∩ 𝑐)) ∧ ((𝑆 ∈ TopBases ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ 𝑣 ∈ (𝑏 ∩ 𝑑))) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
33	32	an4s 647	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ TopBases ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ (𝑆 ∈ TopBases ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) ∧ (𝑢 ∈ (𝑎 ∩ 𝑐) ∧ 𝑣 ∈ (𝑏 ∩ 𝑑))) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
34	33	ralrimivva 3141	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ TopBases ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ (𝑆 ∈ TopBases ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ∀𝑢 ∈ (𝑎 ∩ 𝑐)∀𝑣 ∈ (𝑏 ∩ 𝑑)∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
35		eleq1 2853	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑢, 𝑣⟩ → (𝑝 ∈ (𝑥 × 𝑦) ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦)))
36	35	anbi1d 620	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑢, 𝑣⟩ → ((𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
37	36	2rexbidv 3245	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑢, 𝑣⟩ → (∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
38	37	ralxp 5562	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∀𝑢 ∈ (𝑎 ∩ 𝑐)∀𝑣 ∈ (𝑏 ∩ 𝑑)∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
39	34, 38	sylibr 226	. . . . . . . . . 10 ⊢ (((𝑅 ∈ TopBases ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ (𝑆 ∈ TopBases ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ∀𝑝 ∈ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
40	39	an4s 647	. . . . . . . . 9 ⊢ (((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) ∧ ((𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅) ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ∀𝑝 ∈ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
41	40	anassrs 460	. . . . . . . 8 ⊢ ((((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ∀𝑝 ∈ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
42		ineq12 4071	. . . . . . . . . 10 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (𝑢 ∩ 𝑣) = ((𝑎 × 𝑏) ∩ (𝑐 × 𝑑)))
43		inxp 5553	. . . . . . . . . 10 ⊢ ((𝑎 × 𝑏) ∩ (𝑐 × 𝑑)) = ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))
44	42, 43	syl6eq 2830	. . . . . . . . 9 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (𝑢 ∩ 𝑣) = ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))
45	44	sseq2d 3889	. . . . . . . . . . . 12 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (𝑡 ⊆ (𝑢 ∩ 𝑣) ↔ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
46	45	anbi2d 619	. . . . . . . . . . 11 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → ((𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)) ↔ (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
47	46	rexbidv 3242	. . . . . . . . . 10 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)) ↔ ∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
48	1	rexeqi 3354	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑡 ∈ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
49		fvex 6512	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑧) ∈ V
50		fvex 6512	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝑧) ∈ V
51	49, 50	xpex 7293	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑧) × (2nd ‘𝑧)) ∈ V
52	51	rgenw 3100	. . . . . . . . . . . 12 ⊢ ∀𝑧 ∈ (𝑅 × 𝑆)((1st ‘𝑧) × (2nd ‘𝑧)) ∈ V
53		vex 3418	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
54		vex 3418	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
55	53, 54	op1std 7511	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
56	53, 54	op2ndd 7512	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
57	55, 56	xpeq12d 5438	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) × (2nd ‘𝑧)) = (𝑥 × 𝑦))
58	57	mpompt 7082	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑅 × 𝑆) ↦ ((1st ‘𝑧) × (2nd ‘𝑧))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
59	58	eqcomi 2787	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑧 ∈ (𝑅 × 𝑆) ↦ ((1st ‘𝑧) × (2nd ‘𝑧)))
60		eleq2 2854	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ((1st ‘𝑧) × (2nd ‘𝑧)) → (𝑝 ∈ 𝑡 ↔ 𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧))))
61		sseq1 3882	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ((1st ‘𝑧) × (2nd ‘𝑧)) → (𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)) ↔ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
62	60, 61	anbi12d 621	. . . . . . . . . . . . 13 ⊢ (𝑡 = ((1st ‘𝑧) × (2nd ‘𝑧)) → ((𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ (𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ∧ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
63	59, 62	rexrnmpt 6686	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ (𝑅 × 𝑆)((1st ‘𝑧) × (2nd ‘𝑧)) ∈ V → (∃𝑡 ∈ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑧 ∈ (𝑅 × 𝑆)(𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ∧ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
64	52, 63	ax-mp 5	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑧 ∈ (𝑅 × 𝑆)(𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ∧ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
65	57	eleq2d 2851	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ↔ 𝑝 ∈ (𝑥 × 𝑦)))
66	57	sseq1d 3888	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)) ↔ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
67	65, 66	anbi12d 621	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ∧ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
68	67	rexxp 5563	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ (𝑅 × 𝑆)(𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ∧ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
69	48, 64, 68	3bitri 289	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
70	47, 69	syl6bb 279	. . . . . . . . 9 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)) ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
71	44, 70	raleqbidv 3341	. . . . . . . 8 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)) ↔ ∀𝑝 ∈ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
72	41, 71	syl5ibrcom 239	. . . . . . 7 ⊢ ((((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
73	72	rexlimdvva 3239	. . . . . 6 ⊢ (((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) → (∃𝑏 ∈ 𝑆 ∃𝑑 ∈ 𝑆 (𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
74	17, 73	syl5bir 235	. . . . 5 ⊢ (((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) → ((∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)) → ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
75	74	rexlimdvva 3239	. . . 4 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → (∃𝑎 ∈ 𝑅 ∃𝑐 ∈ 𝑅 (∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)) → ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
76	16, 75	syl5bi 234	. . 3 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
77	76	ralrimivv 3140	. 2 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))
78	1	txbasex 21878	. . 3 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ V)
79		isbasis2g 21260	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
80	78, 79	syl 17	. 2 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → (𝐵 ∈ TopBases ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
81	77, 80	mpbird 249	1 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2758  ∀wral 3088  ∃wrex 3089  Vcvv 3415   ∩ cin 3828   ⊆ wss 3829  ⟨cop 4447   ↦ cmpt 5008   × cxp 5405  ran crn 5408  ‘cfv 6188   ∈ cmpo 6978  1^st c1st 7499  2^nd c2nd 7500  TopBasesctb 21257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-bases 21258

This theorem is referenced by:  txtop  21881  tx2ndc  21963  mbfimaopnlem  23959