Theorem txbasval 14772

Description: It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)

Assertion

Ref	Expression
txbasval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

Proof of Theorem txbasval

Dummy variables 𝑥 𝑦 𝑚 𝑛 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2205	. . . 4 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
2	1	txbasex 14762	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3		bastg 14566	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (topGen‘𝑅))
4		bastg 14566	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ⊆ (topGen‘𝑆))
5		resmpo 6045	. . . . . 6 ⊢ ((𝑅 ⊆ (topGen‘𝑅) ∧ 𝑆 ⊆ (topGen‘𝑆)) → ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) = (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
6	3, 4, 5	syl2an 289	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) = (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
7		resss 4984	. . . . 5 ⊢ ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
8	6, 7	eqsstrrdi 3246	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
9		rnss 4909	. . . 4 ⊢ ((𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
10	8, 9	syl 14	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
11		eltg3 14562	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑢 ∈ (topGen‘𝑅) ↔ ∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚)))
12		eltg3 14562	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑊 → (𝑣 ∈ (topGen‘𝑆) ↔ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
13	11, 12	bi2anan9 606	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅) ∧ 𝑣 ∈ (topGen‘𝑆)) ↔ (∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛))))
14		exdistrv 1934	. . . . . . . . 9 ⊢ (∃𝑚∃𝑛((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ (∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
15		an4 586	. . . . . . . . . . 11 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)))
16		uniiun 3981	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑚 = ∪ 𝑥 ∈ 𝑚 𝑥
17		uniiun 3981	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 = ∪ 𝑦 ∈ 𝑛 𝑦
18	16, 17	xpeq12i 4698	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑚 × ∪ 𝑛) = (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
19		xpiundir 4735	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
20		xpiundi 4734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
21	20	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑚 → (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦))
22	21	iuneq2i 3945	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
23	18, 19, 22	3eqtri 2230	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑚 × ∪ 𝑛) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
24		txvalex 14759	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V)
25	24	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (𝑅 ×t 𝑆) ∈ V)
26	24	ad2antrr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → (𝑅 ×t 𝑆) ∈ V)
27		ssel2 3188	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝑅)
28		ssel2 3188	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ⊆ 𝑆 ∧ 𝑦 ∈ 𝑛) → 𝑦 ∈ 𝑆)
29	27, 28	anim12i 338	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
30	29	an4s 588	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
31		txopn 14770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
32	30, 31	sylan2 286	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛))) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
33	32	anassrs 400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
34	33	anassrs 400	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) ∧ 𝑦 ∈ 𝑛) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
35	34	ralrimiva 2579	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
36		tgiun 14578	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ×t 𝑆) ∈ V ∧ ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆)) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
37	26, 35, 36	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
38		tgidm 14579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V → (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
39	2, 38	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
40	1	txval 14760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
41	40	fveq2d 5582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))))
42	39, 41, 40	3eqtr4d 2248	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
43	42	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
44	43	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
45	37, 44	eleqtrd 2284	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
46	45	ralrimiva 2579	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
47		tgiun 14578	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ×t 𝑆) ∈ V ∧ ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
48	25, 46, 47	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
49	48, 43	eleqtrd 2284	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
50	23, 49	eqeltrid 2292	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆))
51		xpeq12 4695	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) = (∪ 𝑚 × ∪ 𝑛))
52	51	eleq1d 2274	. . . . . . . . . . . . 13 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → ((𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆) ↔ (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆)))
53	50, 52	syl5ibrcom 157	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
54	53	expimpd 363	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
55	15, 54	biimtrid 152	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
56	55	exlimdvv 1921	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∃𝑚∃𝑛((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
57	14, 56	biimtrrid 153	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
58	13, 57	sylbid 150	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅) ∧ 𝑣 ∈ (topGen‘𝑆)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
59	58	ralrimivv 2587	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∀𝑢 ∈ (topGen‘𝑅)∀𝑣 ∈ (topGen‘𝑆)(𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆))
60		eqid 2205	. . . . . . 7 ⊢ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
61	60	fmpo 6289	. . . . . 6 ⊢ (∀𝑢 ∈ (topGen‘𝑅)∀𝑣 ∈ (topGen‘𝑆)(𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆) ↔ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)):((topGen‘𝑅) × (topGen‘𝑆))⟶(𝑅 ×t 𝑆))
62	59, 61	sylib 122	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)):((topGen‘𝑅) × (topGen‘𝑆))⟶(𝑅 ×t 𝑆))
63	62	frnd 5437	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (𝑅 ×t 𝑆))
64	63, 40	sseqtrd 3231	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
65		2basgeng 14587	. . 3 ⊢ ((ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V ∧ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ∧ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))))
66	2, 10, 64, 65	syl3anc 1250	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))))
67		tgvalex 13128	. . 3 ⊢ (𝑅 ∈ 𝑉 → (topGen‘𝑅) ∈ V)
68		tgvalex 13128	. . 3 ⊢ (𝑆 ∈ 𝑊 → (topGen‘𝑆) ∈ V)
69		eqid 2205	. . . 4 ⊢ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
70	69	txval 14760	. . 3 ⊢ (((topGen‘𝑅) ∈ V ∧ (topGen‘𝑆) ∈ V) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))))
71	67, 68, 70	syl2an 289	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))))
72	66, 40, 71	3eqtr4rd 2249	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∀wral 2484  Vcvv 2772   ⊆ wss 3166  ∪ cuni 3850  ∪ ciun 3927   × cxp 4674  ran crn 4677   ↾ cres 4678  ⟶wf 5268  ‘cfv 5272  (class class class)co 5946   ∈ cmpo 5948  topGenctg 13119   ×_t ctx 14757

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-topgen 13125  df-tx 14758

This theorem is referenced by: (None)