Theorem txbasval 23774

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 2762	. . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
2	1	txval 23732	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
3		bastg 23134	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (topGen‘𝑅))
4		bastg 23134	. . . . . . 7 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ⊆ (topGen‘𝑆))
5		resmpo 7532	. . . . . . 7 ⊢ ((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 607	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) = (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
7		resss 5999	. . . . . 6 ⊢ ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
8	6, 7	eqsstrrdi 3981	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
9		rnss 5928	. . . . 5 ⊢ ((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 18	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
11		eltg3 23130	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑢 ∈ (topGen‘𝑅) ↔ ∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚)))
12		eltg3 23130	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑊 → (𝑣 ∈ (topGen‘𝑆) ↔ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
13	11, 12	bi2anan9 649	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅) ∧ 𝑣 ∈ (topGen‘𝑆)) ↔ (∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛))))
14		exdistrv 1984	. . . . . . . . . 10 ⊢ (∃𝑚∃𝑛((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ (∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
15		an4 668	. . . . . . . . . . . 12 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)))
16		uniiun 5022	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑚 = ∪ 𝑥 ∈ 𝑚 𝑥
17		uniiun 5022	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 = ∪ 𝑦 ∈ 𝑛 𝑦
18	16, 17	xpeq12i 5688	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑚 × ∪ 𝑛) = (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
19		xpiundir 5732	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
20		xpiundi 5731	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
21	20	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑚 → (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦))
22	21	iuneq2i 4977	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
23	18, 19, 22	3eqtri 2789	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑚 × ∪ 𝑛) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
24		ovex 7445	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ×t 𝑆) ∈ V
25		ssel2 3931	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝑅)
26		ssel2 3931	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 ⊆ 𝑆 ∧ 𝑦 ∈ 𝑛) → 𝑦 ∈ 𝑆)
27	25, 26	anim12i 624	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
28	27	an4s 672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
29		txopn 23770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
30	28, 29	sylan2 604	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛))) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
31	30	anassrs 472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
32	31	anassrs 472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) ∧ 𝑦 ∈ 𝑛) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
33	32	ralrimiva 3156	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
34		tgiun 23147	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ×t 𝑆) ∈ V ∧ ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆)) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
35	24, 33, 34	sylancr 598	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
36	1	txbasex 23734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
37		tgidm 23148	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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))))
38	36, 37	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
39	2	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))))
40	38, 39, 2	3eqtr4d 2807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
41	40	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
42	41	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
43	35, 42	eleqtrd 2864	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
44	43	ralrimiva 3156	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
45		tgiun 23147	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ×t 𝑆) ∈ V ∧ ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
46	24, 44, 45	sylancr 598	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
47	46, 41	eleqtrd 2864	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
48	23, 47	eqeltrid 2866	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆))
49		xpeq12 5685	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) = (∪ 𝑚 × ∪ 𝑛))
50	49	eleq1d 2847	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → ((𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆) ↔ (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆)))
51	48, 50	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
52	51	expimpd 458	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
53	15, 52	biimtrid 245	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
54	53	exlimdvv 1963	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∃𝑚∃𝑛((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
55	14, 54	biimtrrid 246	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
56	13, 55	sylbid 243	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅) ∧ 𝑣 ∈ (topGen‘𝑆)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
57	56	ralrimivv 3205	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∀𝑢 ∈ (topGen‘𝑅)∀𝑣 ∈ (topGen‘𝑆)(𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆))
58		eqid 2762	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
59	58	fmpo 8063	. . . . . . 7 ⊢ (∀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))
60	57, 59	sylib 221	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)):((topGen‘𝑅) × (topGen‘𝑆))⟶(𝑅 ×t 𝑆))
61	60	frnd 6714	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (𝑅 ×t 𝑆))
62	61, 2	sseqtrd 3972	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
63		2basgen 23158	. . . 4 ⊢ ((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))))
64	10, 62, 63	syl2anc 595	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))))
65		fvex 6894	. . . 4 ⊢ (topGen‘𝑅) ∈ V
66		fvex 6894	. . . 4 ⊢ (topGen‘𝑆) ∈ V
67		eqid 2762	. . . . 5 ⊢ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
68	67	txval 23732	. . . 4 ⊢ (((topGen‘𝑅) ∈ V ∧ (topGen‘𝑆) ∈ V) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))))
69	65, 66, 68	mp2an 704	. . 3 ⊢ ((topGen‘𝑅) ×t (topGen‘𝑆)) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
70	64, 69	eqtr4di 2815	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ((topGen‘𝑅) ×t (topGen‘𝑆)))
71	2, 70	eqtr2d 2798	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ⊆ wss 3904  ∪ cuni 4871  ∪ ciun 4955   × cxp 5658  ran crn 5661   ↾ cres 5662  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  topGenctg 17496   ×_t ctx 23728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-topgen 17502  df-tx 23730

This theorem is used by:  tx2ndc  23819  mbfimaopnlem  25825