Theorem txbasval 14435

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 2193	. . . 4 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
2	1	txbasex 14425	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3		bastg 14229	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (topGen‘𝑅))
4		bastg 14229	. . . . . 6 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ⊆ (topGen‘𝑆))
5		resmpo 6016	. . . . . 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 4966	. . . . 5 ⊢ ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
8	6, 7	eqsstrrdi 3232	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
9		rnss 4892	. . . 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 14225	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑢 ∈ (topGen‘𝑅) ↔ ∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚)))
12		eltg3 14225	. . . . . . . . 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 1922	. . . . . . . . 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 3966	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑚 = ∪ 𝑥 ∈ 𝑚 𝑥
17		uniiun 3966	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 = ∪ 𝑦 ∈ 𝑛 𝑦
18	16, 17	xpeq12i 4681	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑚 × ∪ 𝑛) = (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
19		xpiundir 4718	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
20		xpiundi 4717	. . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
23	18, 19, 22	3eqtri 2218	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑚 × ∪ 𝑛) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
24		txvalex 14422	. . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝑅)
28		ssel2 3174	. . . . . . . . . . . . . . . . . . . . . . . . 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 14433	. . . . . . . . . . . . . . . . . . . . . . 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 2567	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
36		tgiun 14241	. . . . . . . . . . . . . . . . . . 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 14242	. . . . . . . . . . . . . . . . . . . . . 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 14423	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
41	40	fveq2d 5558	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))))
42	39, 41, 40	3eqtr4d 2236	. . . . . . . . . . . . . . . . . . . 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 2272	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
46	45	ralrimiva 2567	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
47		tgiun 14241	. . . . . . . . . . . . . . . 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 2272	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
50	23, 49	eqeltrid 2280	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆))
51		xpeq12 4678	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) = (∪ 𝑚 × ∪ 𝑛))
52	51	eleq1d 2262	. . . . . . . . . . . . 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 1909	. . . . . . . . 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 2575	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∀𝑢 ∈ (topGen‘𝑅)∀𝑣 ∈ (topGen‘𝑆)(𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆))
60		eqid 2193	. . . . . . 7 ⊢ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
61	60	fmpo 6254	. . . . . 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 5413	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (𝑅 ×t 𝑆))
64	63, 40	sseqtrd 3217	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
65		2basgeng 14250	. . 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 1249	. 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 12874	. . 3 ⊢ (𝑅 ∈ 𝑉 → (topGen‘𝑅) ∈ V)
68		tgvalex 12874	. . 3 ⊢ (𝑆 ∈ 𝑊 → (topGen‘𝑆) ∈ V)
69		eqid 2193	. . . 4 ⊢ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
70	69	txval 14423	. . 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 2237	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  Vcvv 2760   ⊆ wss 3153  ∪ cuni 3835  ∪ ciun 3912   × cxp 4657  ran crn 4660   ↾ cres 4661  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  topGenctg 12865   ×_t ctx 14420

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-topgen 12871  df-tx 14421

This theorem is referenced by: (None)