Theorem txbasval 22757

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 2738	. . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
2	1	txval 22715	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
3		bastg 22116	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (topGen‘𝑅))
4		bastg 22116	. . . . . . 7 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ⊆ (topGen‘𝑆))
5		resmpo 7394	. . . . . . 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 596	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) = (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
7		resss 5916	. . . . . 6 ⊢ ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
8	6, 7	eqsstrrdi 3976	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
9		rnss 5848	. . . . 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 17	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
11		eltg3 22112	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑢 ∈ (topGen‘𝑅) ↔ ∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚)))
12		eltg3 22112	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑊 → (𝑣 ∈ (topGen‘𝑆) ↔ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
13	11, 12	bi2anan9 636	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅) ∧ 𝑣 ∈ (topGen‘𝑆)) ↔ (∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛))))
14		exdistrv 1959	. . . . . . . . . 10 ⊢ (∃𝑚∃𝑛((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ (∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
15		an4 653	. . . . . . . . . . . 12 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)))
16		uniiun 4988	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑚 = ∪ 𝑥 ∈ 𝑚 𝑥
17		uniiun 4988	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 = ∪ 𝑦 ∈ 𝑛 𝑦
18	16, 17	xpeq12i 5617	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑚 × ∪ 𝑛) = (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
19		xpiundir 5658	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
20		xpiundi 5657	. . . . . . . . . . . . . . . . . 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 4945	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
23	18, 19, 22	3eqtri 2770	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑚 × ∪ 𝑛) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
24		ovex 7308	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ×t 𝑆) ∈ V
25		ssel2 3916	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝑅)
26		ssel2 3916	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 ⊆ 𝑆 ∧ 𝑦 ∈ 𝑛) → 𝑦 ∈ 𝑆)
27	25, 26	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
28	27	an4s 657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
29		txopn 22753	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
30	28, 29	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛))) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
31	30	anassrs 468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
32	31	anassrs 468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) ∧ 𝑦 ∈ 𝑛) → (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
33	32	ralrimiva 3103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
34		tgiun 22129	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
36	1	txbasex 22717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
37		tgidm 22130	. . . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))))
40	38, 39, 2	3eqtr4d 2788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
41	40	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
42	41	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → (topGen‘(𝑅 ×t 𝑆)) = (𝑅 ×t 𝑆))
43	35, 42	eleqtrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
44	43	ralrimiva 3103	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
45		tgiun 22129	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (topGen‘(𝑅 ×t 𝑆)))
47	46, 41	eleqtrd 2841	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
48	23, 47	eqeltrid 2843	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆))
49		xpeq12 5614	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) = (∪ 𝑚 × ∪ 𝑛))
50	49	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → ((𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆) ↔ (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆)))
51	48, 50	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
52	51	expimpd 454	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
53	15, 52	syl5bi 241	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
54	53	exlimdvv 1937	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∃𝑚∃𝑛((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
55	14, 54	syl5bir 242	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
56	13, 55	sylbid 239	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((𝑢 ∈ (topGen‘𝑅) ∧ 𝑣 ∈ (topGen‘𝑆)) → (𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆)))
57	56	ralrimivv 3122	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∀𝑢 ∈ (topGen‘𝑅)∀𝑣 ∈ (topGen‘𝑆)(𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆))
58		eqid 2738	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
59	58	fmpo 7908	. . . . . . 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 217	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)):((topGen‘𝑅) × (topGen‘𝑆))⟶(𝑅 ×t 𝑆))
61	60	frnd 6608	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (𝑅 ×t 𝑆))
62	61, 2	sseqtrd 3961	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
63		2basgen 22140	. . . 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 584	. . 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 6787	. . . 4 ⊢ (topGen‘𝑅) ∈ V
66		fvex 6787	. . . 4 ⊢ (topGen‘𝑆) ∈ V
67		eqid 2738	. . . . 5 ⊢ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
68	67	txval 22715	. . . 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 689	. . 3 ⊢ ((topGen‘𝑅) ×t (topGen‘𝑆)) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
70	64, 69	eqtr4di 2796	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ((topGen‘𝑅) ×t (topGen‘𝑆)))
71	2, 70	eqtr2d 2779	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ∪ ciun 4924   × cxp 5587  ran crn 5590   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  topGenctg 17148   ×_t ctx 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-topgen 17154  df-tx 22713

This theorem is referenced by:  tx2ndc  22802  mbfimaopnlem  24819