Theorem xkopt 22260

Description: The compact-open topology on a discrete set coincides with the product topology where all the factors are the same. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
xkopt	⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑅 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))

Proof of Theorem xkopt

Dummy variables 𝑓 𝑘 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 21600	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		simpl 486	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ Top)
3		unipw 5308	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
4	3	eqcomi 2807	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
5		eqid 2798	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}
6		eqid 2798	. . . . 5 ⊢ (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
7	4, 5, 6	xkoval 22192	. . . 4 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ↑ko 𝒫 𝐴) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
8	1, 2, 7	syl2an2 685	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑅 ↑ko 𝒫 𝐴) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
9		simpr 488	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
10		fconst6g 6542	. . . . . 6 ⊢ (𝑅 ∈ Top → (𝐴 × {𝑅}):𝐴⟶Top)
11	10	adantr 484	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 × {𝑅}):𝐴⟶Top)
12		pttop 22187	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 × {𝑅}):𝐴⟶Top) → (∏t‘(𝐴 × {𝑅})) ∈ Top)
13	9, 11, 12	syl2anc 587	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∏t‘(𝐴 × {𝑅})) ∈ Top)
14		elpwi 4506	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
15		restdis 21783	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ⊆ 𝐴) → (𝒫 𝐴 ↾t 𝑥) = 𝒫 𝑥)
16	14, 15	sylan2 595	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝒫 𝐴 ↾t 𝑥) = 𝒫 𝑥)
17	16	adantll 713	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 𝐴) → (𝒫 𝐴 ↾t 𝑥) = 𝒫 𝑥)
18	17	eleq1d 2874	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝒫 𝐴 ↾t 𝑥) ∈ Comp ↔ 𝒫 𝑥 ∈ Comp))
19		discmp 22003	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Comp)
20	18, 19	syl6bbr 292	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝒫 𝐴 ↾t 𝑥) ∈ Comp ↔ 𝑥 ∈ Fin))
21	20	rabbidva 3425	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ∈ Fin})
22		dfin5 3889	. . . . . . . . 9 ⊢ (𝒫 𝐴 ∩ Fin) = {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ∈ Fin}
23	21, 22	eqtr4di 2851	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp} = (𝒫 𝐴 ∩ Fin))
24		eqidd 2799	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑅)
25		toptopon2 21523	. . . . . . . . . 10 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
26		cndis 21896	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘∪ 𝑅)) → (𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑m 𝐴))
27	26	ancoms 462	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑m 𝐴))
28	25, 27	sylanb 584	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑m 𝐴))
29	28	rabeqdv 3432	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
30	23, 24, 29	mpoeq123dv 7208	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ (𝒫 𝐴 ∩ Fin), 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
31	30	rneqd 5772	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = ran (𝑘 ∈ (𝒫 𝐴 ∩ Fin), 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
32		eqid 2798	. . . . . . 7 ⊢ (𝑘 ∈ (𝒫 𝐴 ∩ Fin), 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ (𝒫 𝐴 ∩ Fin), 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
33	32	rnmpo 7263	. . . . . 6 ⊢ ran (𝑘 ∈ (𝒫 𝐴 ∩ Fin), 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑥 ∣ ∃𝑘 ∈ (𝒫 𝐴 ∩ Fin)∃𝑣 ∈ 𝑅 𝑥 = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
34	31, 33	eqtrdi 2849	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑥 ∣ ∃𝑘 ∈ (𝒫 𝐴 ∩ Fin)∃𝑣 ∈ 𝑅 𝑥 = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}})
35		elmapi 8411	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (∪ 𝑅 ↑m 𝐴) → 𝑓:𝐴⟶∪ 𝑅)
36		eleq2 2878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑓‘𝑥) ∈ 𝑣 ↔ (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
37	36	imbi2d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝑣) ↔ (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))))
38	37	bibi1d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → (((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝑣) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣)) ↔ ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣))))
39		eleq2 2878	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑓‘𝑥) ∈ ∪ 𝑅 ↔ (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
40	39	imbi2d 344	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅) ↔ (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))))
41	40	bibi1d 347	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → (((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣)) ↔ ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣))))
42		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ∈ (𝒫 𝐴 ∩ Fin))
43	42	elin1d 4125	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ∈ 𝒫 𝐴)
44	43	elpwid 4508	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ⊆ 𝐴)
45	44	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → 𝑘 ⊆ 𝐴)
46	45	sselda 3915	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝐴)
47		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝑘)
48	46, 47	2thd 268	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝑘))
49	48	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝑣) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣)))
50		ffvelrn 6826	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴⟶∪ 𝑅 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ∪ 𝑅)
51	50	ex 416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴⟶∪ 𝑅 → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅))
52	51	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅))
53	52	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ ¬ 𝑥 ∈ 𝑘) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅))
54		pm2.21 123	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ 𝑘 → (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣))
55	54	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ ¬ 𝑥 ∈ 𝑘) → (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣))
56	53, 55	2thd 268	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ ¬ 𝑥 ∈ 𝑘) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣)))
57	38, 41, 49, 56	ifbothda 4462	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)) ↔ (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣)))
58	57	ralbidv2 3160	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ ∀𝑥 ∈ 𝑘 (𝑓‘𝑥) ∈ 𝑣))
59		ffn 6487	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴⟶∪ 𝑅 → 𝑓 Fn 𝐴)
60	59	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → 𝑓 Fn 𝐴)
61		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
62	61	elixp 8451	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
63	62	baib 539	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝐴 → (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
64	60, 63	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
65		ffun 6490	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴⟶∪ 𝑅 → Fun 𝑓)
66		fdm 6495	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴⟶∪ 𝑅 → dom 𝑓 = 𝐴)
67	66	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → dom 𝑓 = 𝐴)
68	45, 67	sseqtrrd 3956	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → 𝑘 ⊆ dom 𝑓)
69		funimass4 6705	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝑓 ∧ 𝑘 ⊆ dom 𝑓) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑘 (𝑓‘𝑥) ∈ 𝑣))
70	65, 68, 69	syl2an2 685	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑘 (𝑓‘𝑥) ∈ 𝑣))
71	58, 64, 70	3bitr4d 314	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ (𝑓 “ 𝑘) ⊆ 𝑣))
72	35, 71	sylan2 595	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓 ∈ (∪ 𝑅 ↑m 𝐴)) → (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ (𝑓 “ 𝑘) ⊆ 𝑣))
73	72	rabbi2dva 4144	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → ((∪ 𝑅 ↑m 𝐴) ∩ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)) = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
74		elssuni 4830	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑅 → 𝑣 ⊆ ∪ 𝑅)
75	74	ad2antll 728	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑣 ⊆ ∪ 𝑅)
76		ssid 3937	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑅 ⊆ ∪ 𝑅
77		sseq1 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → (𝑣 ⊆ ∪ 𝑅 ↔ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅))
78		sseq1 3940	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → (∪ 𝑅 ⊆ ∪ 𝑅 ↔ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅))
79	77, 78	ifboth 4463	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ⊆ ∪ 𝑅 ∧ ∪ 𝑅 ⊆ ∪ 𝑅) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅)
80	75, 76, 79	sylancl 589	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅)
81	80	ralrimivw 3150	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → ∀𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅)
82		ss2ixp 8457	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅 → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ X𝑥 ∈ 𝐴 ∪ 𝑅)
83	81, 82	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ X𝑥 ∈ 𝐴 ∪ 𝑅)
84		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝐴 ∈ 𝑉)
85		uniexg 7446	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ V)
86	85	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → ∪ 𝑅 ∈ V)
87		ixpconstg 8453	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑅 ∈ V) → X𝑥 ∈ 𝐴 ∪ 𝑅 = (∪ 𝑅 ↑m 𝐴))
88	84, 86, 87	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 ∪ 𝑅 = (∪ 𝑅 ↑m 𝐴))
89	83, 88	sseqtrd 3955	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ (∪ 𝑅 ↑m 𝐴))
90		sseqin2 4142	. . . . . . . . . . 11 ⊢ (X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ (∪ 𝑅 ↑m 𝐴) ↔ ((∪ 𝑅 ↑m 𝐴) ∩ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)) = X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))
91	89, 90	sylib 221	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → ((∪ 𝑅 ↑m 𝐴) ∩ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)) = X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))
92	73, 91	eqtr3d 2835	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))
93	10	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → (𝐴 × {𝑅}):𝐴⟶Top)
94	42	elin2d 4126	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ∈ Fin)
95		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → 𝑣 ∈ 𝑅)
96		eqid 2798	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑅 = ∪ 𝑅
97	96	topopn 21511	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
98	97	ad3antrrr 729	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → ∪ 𝑅 ∈ 𝑅)
99	95, 98	ifcld 4470	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ∈ 𝑅)
100		fvconst2g 6941	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑅})‘𝑥) = 𝑅)
101	100	ad4ant14 751	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑅})‘𝑥) = 𝑅)
102	99, 101	eleqtrrd 2893	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ∈ ((𝐴 × {𝑅})‘𝑥))
103		eldifn 4055	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝑘) → ¬ 𝑥 ∈ 𝑘)
104	103	iffalsed 4436	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝑘) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) = ∪ 𝑅)
105	104	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) = ∪ 𝑅)
106		eldifi 4054	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝑘) → 𝑥 ∈ 𝐴)
107	106, 101	sylan2 595	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → ((𝐴 × {𝑅})‘𝑥) = 𝑅)
108	107	unieqd 4814	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → ∪ ((𝐴 × {𝑅})‘𝑥) = ∪ 𝑅)
109	105, 108	eqtr4d 2836	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) = ∪ ((𝐴 × {𝑅})‘𝑥))
110	84, 93, 94, 102, 109	ptopn 22188	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ∈ (∏t‘(𝐴 × {𝑅})))
111	92, 110	eqeltrd 2890	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ (∏t‘(𝐴 × {𝑅})))
112		eleq1 2877	. . . . . . . 8 ⊢ (𝑥 = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (𝑥 ∈ (∏t‘(𝐴 × {𝑅})) ↔ {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ (∏t‘(𝐴 × {𝑅}))))
113	111, 112	syl5ibrcom 250	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → (𝑥 = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → 𝑥 ∈ (∏t‘(𝐴 × {𝑅}))))
114	113	rexlimdvva 3253	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∃𝑘 ∈ (𝒫 𝐴 ∩ Fin)∃𝑣 ∈ 𝑅 𝑥 = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → 𝑥 ∈ (∏t‘(𝐴 × {𝑅}))))
115	114	abssdv 3996	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑥 ∣ ∃𝑘 ∈ (𝒫 𝐴 ∩ Fin)∃𝑣 ∈ 𝑅 𝑥 = {𝑓 ∈ (∪ 𝑅 ↑m 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ⊆ (∏t‘(𝐴 × {𝑅})))
116	34, 115	eqsstrd 3953	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (∏t‘(𝐴 × {𝑅})))
117		tgfiss 21596	. . . 4 ⊢ (((∏t‘(𝐴 × {𝑅})) ∈ Top ∧ ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ (∏t‘(𝐴 × {𝑅}))) → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) ⊆ (∏t‘(𝐴 × {𝑅})))
118	13, 116, 117	syl2anc 587	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑅 ↦ {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) ⊆ (∏t‘(𝐴 × {𝑅})))
119	8, 118	eqsstrd 3953	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑅 ↑ko 𝒫 𝐴) ⊆ (∏t‘(𝐴 × {𝑅})))
120		eqid 2798	. . . . . . . 8 ⊢ (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝐴 × {𝑅}))
121	120, 96	ptuniconst 22203	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → (∪ 𝑅 ↑m 𝐴) = ∪ (∏t‘(𝐴 × {𝑅})))
122	121	ancoms 462	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∪ 𝑅 ↑m 𝐴) = ∪ (∏t‘(𝐴 × {𝑅})))
123	28, 122	eqtrd 2833	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 Cn 𝑅) = ∪ (∏t‘(𝐴 × {𝑅})))
124	123	oveq2d 7151	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) = ((∏t‘(𝐴 × {𝑅})) ↾t ∪ (∏t‘(𝐴 × {𝑅}))))
125		eqid 2798	. . . . . 6 ⊢ ∪ (∏t‘(𝐴 × {𝑅})) = ∪ (∏t‘(𝐴 × {𝑅}))
126	125	restid 16699	. . . . 5 ⊢ ((∏t‘(𝐴 × {𝑅})) ∈ Top → ((∏t‘(𝐴 × {𝑅})) ↾t ∪ (∏t‘(𝐴 × {𝑅}))) = (∏t‘(𝐴 × {𝑅})))
127	13, 126	syl 17	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t ∪ (∏t‘(𝐴 × {𝑅}))) = (∏t‘(𝐴 × {𝑅})))
128	124, 127	eqtrd 2833	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) = (∏t‘(𝐴 × {𝑅})))
129	4, 120	xkoptsub 22259	. . . 4 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝑅 ∈ Top) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) ⊆ (𝑅 ↑ko 𝒫 𝐴))
130	1, 2, 129	syl2an2 685	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) ⊆ (𝑅 ↑ko 𝒫 𝐴))
131	128, 130	eqsstrrd 3954	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∏t‘(𝐴 × {𝑅})) ⊆ (𝑅 ↑ko 𝒫 𝐴))
132	119, 131	eqssd 3932	1 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑅 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ifcif 4425  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800   × cxp 5517  dom cdm 5519  ran crn 5520   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389  Xcixp 8444  Fincfn 8492  ficfi 8858   ↾_t crest 16686  topGenctg 16703  ∏_tcpt 16704  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  Compccmp 21991   ↑_ko cxko 22166

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-pt 16710  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-cmp 21992  df-xko 22168

This theorem is referenced by:  tmdgsum  22700  tmdgsum2  22701  efmndtmd  22706  symgtgp  22711