Theorem txbasval 22957

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 2736	. . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
2	1	txval 22915	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
3		bastg 22316	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (topGen‘𝑅))
4		bastg 22316	. . . . . . 7 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ⊆ (topGen‘𝑆))
5		resmpo 7476	. . . . . . 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 5962	. . . . . 6 ⊢ ((𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ↾ (𝑅 × 𝑆)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
8	6, 7	eqsstrrdi 3999	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
9		rnss 5894	. . . . 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 22312	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (𝑢 ∈ (topGen‘𝑅) ↔ ∃𝑚(𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚)))
12		eltg3 22312	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑊 → (𝑣 ∈ (topGen‘𝑆) ↔ ∃𝑛(𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)))
13	11, 12	bi2anan9 637	. . . . . . . . 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 654	. . . . . . . . . . . 12 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑢 = ∪ 𝑚) ∧ (𝑛 ⊆ 𝑆 ∧ 𝑣 = ∪ 𝑛)) ↔ ((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛)))
16		uniiun 5018	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑚 = ∪ 𝑥 ∈ 𝑚 𝑥
17		uniiun 5018	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 = ∪ 𝑦 ∈ 𝑛 𝑦
18	16, 17	xpeq12i 5661	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑚 × ∪ 𝑛) = (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
19		xpiundir 5703	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑥 ∈ 𝑚 𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦)
20		xpiundi 5702	. . . . . . . . . . . . . . . . . 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 4975	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ 𝑚 (𝑥 × ∪ 𝑦 ∈ 𝑛 𝑦) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
23	18, 19, 22	3eqtri 2768	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑚 × ∪ 𝑛) = ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦)
24		ovex 7390	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ×t 𝑆) ∈ V
25		ssel2 3939	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑅 ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝑅)
26		ssel2 3939	. . . . . . . . . . . . . . . . . . . . . . . . . 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 658	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛)) → (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆))
29		txopn 22953	. . . . . . . . . . . . . . . . . . . . . . . 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 3143	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∀𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
34		tgiun 22329	. . . . . . . . . . . . . . . . . . . 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 22917	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
37		tgidm 22330	. . . . . . . . . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘(𝑅 ×t 𝑆)) = (topGen‘(topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))))
40	38, 39, 2	3eqtr4d 2786	. . . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) ∧ 𝑥 ∈ 𝑚) → ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
44	43	ralrimiva 3143	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∀𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
45		tgiun 22329	. . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → ∪ 𝑥 ∈ 𝑚 ∪ 𝑦 ∈ 𝑛 (𝑥 × 𝑦) ∈ (𝑅 ×t 𝑆))
48	23, 47	eqeltrid 2842	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑚 ⊆ 𝑅 ∧ 𝑛 ⊆ 𝑆)) → (∪ 𝑚 × ∪ 𝑛) ∈ (𝑅 ×t 𝑆))
49		xpeq12 5658	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = ∪ 𝑚 ∧ 𝑣 = ∪ 𝑛) → (𝑢 × 𝑣) = (∪ 𝑚 × ∪ 𝑛))
50	49	eleq1d 2822	. . . . . . . . . . . . . 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	biimtrid 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	biimtrrid 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 3195	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∀𝑢 ∈ (topGen‘𝑅)∀𝑣 ∈ (topGen‘𝑆)(𝑢 × 𝑣) ∈ (𝑅 ×t 𝑆))
58		eqid 2736	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
59	58	fmpo 8000	. . . . . . 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 6676	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (𝑅 ×t 𝑆))
62	61, 2	sseqtrd 3984	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
63		2basgen 22340	. . . 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 6855	. . . 4 ⊢ (topGen‘𝑅) ∈ V
66		fvex 6855	. . . 4 ⊢ (topGen‘𝑆) ∈ V
67		eqid 2736	. . . . 5 ⊢ ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣))
68	67	txval 22915	. . . 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 690	. . 3 ⊢ ((topGen‘𝑅) ×t (topGen‘𝑆)) = (topGen‘ran (𝑢 ∈ (topGen‘𝑅), 𝑣 ∈ (topGen‘𝑆) ↦ (𝑢 × 𝑣)))
70	64, 69	eqtr4di 2794	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ((topGen‘𝑅) ×t (topGen‘𝑆)))
71	2, 70	eqtr2d 2777	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ⊆ wss 3910  ∪ cuni 4865  ∪ ciun 4954   × cxp 5631  ran crn 5634   ↾ cres 5635  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  topGenctg 17319   ×_t ctx 22911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-topgen 17325  df-tx 22913

This theorem is referenced by:  tx2ndc  23002  mbfimaopnlem  25019