Theorem txbas 14494

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 4677	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑦) = (𝑎 × 𝑦))
3		xpeq2 4678	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (𝑎 × 𝑦) = (𝑎 × 𝑏))
4	2, 3	cbvmpov 6002	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑎 ∈ 𝑅, 𝑏 ∈ 𝑆 ↦ (𝑎 × 𝑏))
5	4	rnmpo 6033	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑢 ∣ ∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏)}
6	1, 5	eqtri 2217	. . . . . . 7 ⊢ 𝐵 = {𝑢 ∣ ∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏)}
7	6	abeq2i 2307	. . . . . 6 ⊢ (𝑢 ∈ 𝐵 ↔ ∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏))
8		xpeq1 4677	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → (𝑥 × 𝑦) = (𝑐 × 𝑦))
9		xpeq2 4678	. . . . . . . . . 10 ⊢ (𝑦 = 𝑑 → (𝑐 × 𝑦) = (𝑐 × 𝑑))
10	8, 9	cbvmpov 6002	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑐 ∈ 𝑅, 𝑑 ∈ 𝑆 ↦ (𝑐 × 𝑑))
11	10	rnmpo 6033	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑣 ∣ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)}
12	1, 11	eqtri 2217	. . . . . . 7 ⊢ 𝐵 = {𝑣 ∣ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)}
13	12	abeq2i 2307	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↔ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑))
14	7, 13	anbi12i 460	. . . . 5 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ↔ (∃𝑎 ∈ 𝑅 ∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑐 ∈ 𝑅 ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
15		reeanv 2667	. . . . 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 187	. . . 4 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ↔ ∃𝑎 ∈ 𝑅 ∃𝑐 ∈ 𝑅 (∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
17		reeanv 2667	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑆 ∃𝑑 ∈ 𝑆 (𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) ↔ (∃𝑏 ∈ 𝑆 𝑢 = (𝑎 × 𝑏) ∧ ∃𝑑 ∈ 𝑆 𝑣 = (𝑐 × 𝑑)))
18		basis2 14284	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ TopBases ∧ 𝑎 ∈ 𝑅) ∧ (𝑐 ∈ 𝑅 ∧ 𝑢 ∈ (𝑎 ∩ 𝑐))) → ∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)))
19	18	exp43 372	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ TopBases → (𝑎 ∈ 𝑅 → (𝑐 ∈ 𝑅 → (𝑢 ∈ (𝑎 ∩ 𝑐) → ∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐))))))
20	19	imp42 354	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ TopBases ∧ (𝑎 ∈ 𝑅 ∧ 𝑐 ∈ 𝑅)) ∧ 𝑢 ∈ (𝑎 ∩ 𝑐)) → ∃𝑥 ∈ 𝑅 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)))
21		basis2 14284	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ TopBases ∧ 𝑏 ∈ 𝑆) ∧ (𝑑 ∈ 𝑆 ∧ 𝑣 ∈ (𝑏 ∩ 𝑑))) → ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑)))
22	21	exp43 372	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ TopBases → (𝑏 ∈ 𝑆 → (𝑑 ∈ 𝑆 → (𝑣 ∈ (𝑏 ∩ 𝑑) → ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))))))
23	22	imp42 354	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ TopBases ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) ∧ 𝑣 ∈ (𝑏 ∩ 𝑑)) → ∃𝑦 ∈ 𝑆 (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑)))
24		reeanv 2667	. . . . . . . . . . . . . . 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 4695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) → ⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦))
26		xpss12 4770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ (𝑎 ∩ 𝑐) ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑)) → (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))
27	25, 26	anim12i 338	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑦) ∧ (𝑥 ⊆ (𝑎 ∩ 𝑐) ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
28	27	an4s 588	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑎 ∩ 𝑐)) ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑏 ∩ 𝑑))) → (⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
29	28	reximi 2594	. . . . . . . . . . . . . . . 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 2594	. . . . . . . . . . . . . . 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 135	. . . . . . . . . . . . . 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 289	. . . . . . . . . . . . 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 588	. . . . . . . . . . . 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 2579	. . . . . . . . . . 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 2259	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨𝑢, 𝑣⟩ → (𝑝 ∈ (𝑥 × 𝑦) ↔ ⟨𝑢, 𝑣⟩ ∈ (𝑥 × 𝑦)))
36	35	anbi1d 465	. . . . . . . . . . . . 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 2522	. . . . . . . . . . . 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 4809	. . . . . . . . . . 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 134	. . . . . . . . . 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 588	. . . . . . . . 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 400	. . . . . . . 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 3359	. . . . . . . . . 10 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (𝑢 ∩ 𝑣) = ((𝑎 × 𝑏) ∩ (𝑐 × 𝑑)))
43		inxp 4800	. . . . . . . . . 10 ⊢ ((𝑎 × 𝑏) ∩ (𝑐 × 𝑑)) = ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))
44	42, 43	eqtrdi 2245	. . . . . . . . 9 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (𝑢 ∩ 𝑣) = ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))
45	44	sseq2d 3213	. . . . . . . . . . . 12 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → (𝑡 ⊆ (𝑢 ∩ 𝑣) ↔ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
46	45	anbi2d 464	. . . . . . . . . . 11 ⊢ ((𝑢 = (𝑎 × 𝑏) ∧ 𝑣 = (𝑐 × 𝑑)) → ((𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)) ↔ (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)))))
47	46	rexbidv 2498	. . . . . . . . . 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 2698	. . . . . . . . . . 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		1stexg 6225	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ V → (1st ‘𝑧) ∈ V)
50	49	elv 2767	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑧) ∈ V
51		2ndexg 6226	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ V → (2nd ‘𝑧) ∈ V)
52	51	elv 2767	. . . . . . . . . . . . . 14 ⊢ (2nd ‘𝑧) ∈ V
53	50, 52	xpex 4778	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑧) × (2nd ‘𝑧)) ∈ V
54	53	rgenw 2552	. . . . . . . . . . . 12 ⊢ ∀𝑧 ∈ (𝑅 × 𝑆)((1st ‘𝑧) × (2nd ‘𝑧)) ∈ V
55		vex 2766	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
56		vex 2766	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
57	55, 56	op1std 6206	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
58	55, 56	op2ndd 6207	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
59	57, 58	xpeq12d 4688	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) × (2nd ‘𝑧)) = (𝑥 × 𝑦))
60	59	mpompt 6014	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑅 × 𝑆) ↦ ((1st ‘𝑧) × (2nd ‘𝑧))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
61	60	eqcomi 2200	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑧 ∈ (𝑅 × 𝑆) ↦ ((1st ‘𝑧) × (2nd ‘𝑧)))
62		eleq2 2260	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ((1st ‘𝑧) × (2nd ‘𝑧)) → (𝑝 ∈ 𝑡 ↔ 𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧))))
63		sseq1 3206	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ((1st ‘𝑧) × (2nd ‘𝑧)) → (𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)) ↔ ((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
64	62, 63	anbi12d 473	. . . . . . . . . . . . 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)))))
65	61, 64	rexrnmpt 5705	. . . . . . . . . . . 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)))))
66	54, 65	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))))
67	59	eleq2d 2266	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑝 ∈ ((1st ‘𝑧) × (2nd ‘𝑧)) ↔ 𝑝 ∈ (𝑥 × 𝑦)))
68	59	sseq1d 3212	. . . . . . . . . . . . 13 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((1st ‘𝑧) × (2nd ‘𝑧)) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑)) ↔ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
69	67, 68	anbi12d 473	. . . . . . . . . . . 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)))))
70	69	rexxp 4810	. . . . . . . . . . 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))))
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))) ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ ((𝑎 ∩ 𝑐) × (𝑏 ∩ 𝑑))))
72	47, 71	bitrdi 196	. . . . . . . . 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)))))
73	44, 72	raleqbidv 2709	. . . . . . . 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)))))
74	41, 73	syl5ibrcom 157	. . . . . . 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))))
75	74	rexlimdvva 2622	. . . . . 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))))
76	17, 75	biimtrrid 153	. . . . 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))))
77	76	rexlimdvva 2622	. . . 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))))
78	16, 77	biimtrid 152	. . 3 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
79	78	ralrimivv 2578	. 2 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))
80	1	txbasex 14493	. . 3 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ V)
81		isbasis2g 14281	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
82	80, 81	syl 14	. 2 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → (𝐵 ∈ TopBases ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑝 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑝 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
83	79, 82	mpbird 167	1 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  ⟨cop 3625   ↦ cmpt 4094   × cxp 4661  ran crn 4664  ‘cfv 5258   ∈ cmpo 5924  1^st c1st 6196  2^nd c2nd 6197  TopBasesctb 14278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-bases 14279

This theorem is referenced by:  txtop  14496