Theorem xkopt 21458

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 20799	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2	1	adantl 482	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝒫 𝐴 ∈ Top)
3		simpl 473	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ Top)
4		unipw 4918	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
5	4	eqcomi 2631	. . . . 5 ⊢ 𝐴 = ∪ 𝒫 𝐴
6		eqid 2622	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp}
7		eqid 2622	. . . . 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})
8	5, 6, 7	xkoval 21390	. . . 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}))))
9	2, 3, 8	syl2anc 693	. . 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}))))
10		simpr 477	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
11		fconst6g 6094	. . . . . 6 ⊢ (𝑅 ∈ Top → (𝐴 × {𝑅}):𝐴⟶Top)
12	11	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 × {𝑅}):𝐴⟶Top)
13		pttop 21385	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 × {𝑅}):𝐴⟶Top) → (∏t‘(𝐴 × {𝑅})) ∈ Top)
14	10, 12, 13	syl2anc 693	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∏t‘(𝐴 × {𝑅})) ∈ Top)
15		elpwi 4168	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
16		restdis 20982	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ⊆ 𝐴) → (𝒫 𝐴 ↾t 𝑥) = 𝒫 𝑥)
17	15, 16	sylan2 491	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝒫 𝐴 ↾t 𝑥) = 𝒫 𝑥)
18	17	adantll 750	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 𝐴) → (𝒫 𝐴 ↾t 𝑥) = 𝒫 𝑥)
19	18	eleq1d 2686	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝒫 𝐴 ↾t 𝑥) ∈ Comp ↔ 𝒫 𝑥 ∈ Comp))
20		discmp 21201	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Comp)
21	19, 20	syl6bbr 278	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝒫 𝐴 ↾t 𝑥) ∈ Comp ↔ 𝑥 ∈ Fin))
22	21	rabbidva 3188	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ∈ Fin})
23		dfin5 3582	. . . . . . . . 9 ⊢ (𝒫 𝐴 ∩ Fin) = {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ∈ Fin}
24	22, 23	syl6eqr 2674	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝒫 𝐴 ↾t 𝑥) ∈ Comp} = (𝒫 𝐴 ∩ Fin))
25		eqidd 2623	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑅)
26		eqid 2622	. . . . . . . . . . 11 ⊢ ∪ 𝑅 = ∪ 𝑅
27	26	toptopon 20722	. . . . . . . . . 10 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
28		cndis 21095	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘∪ 𝑅)) → (𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑𝑚 𝐴))
29	28	ancoms 469	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑𝑚 𝐴))
30	27, 29	sylanb 489	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑𝑚 𝐴))
31		rabeq 3192	. . . . . . . . 9 ⊢ ((𝒫 𝐴 Cn 𝑅) = (∪ 𝑅 ↑𝑚 𝐴) → {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑓 ∈ (𝒫 𝐴 Cn 𝑅) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
33	24, 25, 32	mpt2eq123dv 6717	. . . . . . 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}))
34	33	rneqd 5353	. . . . . 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}))
35		eqid 2622	. . . . . . 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})
36	35	rnmpt2 6770	. . . . . 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}}
37	34, 36	syl6eq 2672	. . . . 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}})
38		elmapi 7879	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) → 𝑓:𝐴⟶∪ 𝑅)
39		eleq2 2690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑓‘𝑥) ∈ 𝑣 ↔ (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
40	39	imbi2d 330	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝑣) ↔ (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))))
41	40	bibi1d 333	. . . . . . . . . . . . . . 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))))
42		eleq2 2690	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑓‘𝑥) ∈ ∪ 𝑅 ↔ (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
43	42	imbi2d 330	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅) ↔ (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅))))
44	43	bibi1d 333	. . . . . . . . . . . . . . 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))))
45		inss1 3833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
46		simprl 794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ∈ (𝒫 𝐴 ∩ Fin))
47	45, 46	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ∈ 𝒫 𝐴)
48	47	elpwid 4170	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ⊆ 𝐴)
49	48	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → 𝑘 ⊆ 𝐴)
50	49	sselda 3603	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝐴)
51		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝑘)
52	50, 51	2thd 255	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ 𝑥 ∈ 𝑘) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝑘))
53	52	imbi1d 331	. . . . . . . . . . . . . . 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)))
54		ffvelrn 6357	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴⟶∪ 𝑅 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ∪ 𝑅)
55	54	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴⟶∪ 𝑅 → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅))
56	55	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅))
57	56	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ ¬ 𝑥 ∈ 𝑘) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ ∪ 𝑅))
58		pm2.21 120	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ 𝑘 → (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣))
59	58	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) ∧ ¬ 𝑥 ∈ 𝑘) → (𝑥 ∈ 𝑘 → (𝑓‘𝑥) ∈ 𝑣))
60	57, 59	2thd 255	. . . . . . . . . . . . . . 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)))
61	41, 44, 53, 60	ifbothda 4123	. . . . . . . . . . . . . 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)))
62	61	ralbidv2 2984	. . . . . . . . . . . . 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))
63		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴⟶∪ 𝑅 → 𝑓 Fn 𝐴)
64	63	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → 𝑓 Fn 𝐴)
65		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
66	65	elixp 7915	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
67	66	baib 944	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝐴 → (𝑓 ∈ X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅)))
68	64, 67	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)))
69		ffun 6048	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴⟶∪ 𝑅 → Fun 𝑓)
70	69	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → Fun 𝑓)
71		fdm 6051	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴⟶∪ 𝑅 → dom 𝑓 = 𝐴)
72	71	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → dom 𝑓 = 𝐴)
73	49, 72	sseqtr4d 3642	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → 𝑘 ⊆ dom 𝑓)
74		funimass4 6247	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝑓 ∧ 𝑘 ⊆ dom 𝑓) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑘 (𝑓‘𝑥) ∈ 𝑣))
75	70, 73, 74	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑓:𝐴⟶∪ 𝑅) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑘 (𝑓‘𝑥) ∈ 𝑣))
76	62, 68, 75	3bitr4d 300	. . . . . . . . . . . 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))
77	38, 76	sylan2 491	. . . . . . . . . . 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))
78	77	rabbi2dva 3821	. . . . . . . . . 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})
79		elssuni 4467	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ 𝑅 → 𝑣 ⊆ ∪ 𝑅)
80	79	ad2antll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑣 ⊆ ∪ 𝑅)
81		ssid 3624	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑅 ⊆ ∪ 𝑅
82		sseq1 3626	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → (𝑣 ⊆ ∪ 𝑅 ↔ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅))
83		sseq1 3626	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑅 = if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) → (∪ 𝑅 ⊆ ∪ 𝑅 ↔ if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅))
84	82, 83	ifboth 4124	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ⊆ ∪ 𝑅 ∧ ∪ 𝑅 ⊆ ∪ 𝑅) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅)
85	80, 81, 84	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅)
86	85	ralrimivw 2967	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → ∀𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅)
87		ss2ixp 7921	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ ∪ 𝑅 → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ X𝑥 ∈ 𝐴 ∪ 𝑅)
88	86, 87	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ X𝑥 ∈ 𝐴 ∪ 𝑅)
89		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝐴 ∈ 𝑉)
90		uniexg 6955	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ V)
91	90	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → ∪ 𝑅 ∈ V)
92		ixpconstg 7917	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑅 ∈ V) → X𝑥 ∈ 𝐴 ∪ 𝑅 = (∪ 𝑅 ↑𝑚 𝐴))
93	89, 91, 92	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 ∪ 𝑅 = (∪ 𝑅 ↑𝑚 𝐴))
94	88, 93	sseqtrd 3641	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ⊆ (∪ 𝑅 ↑𝑚 𝐴))
95		sseqin2 3817	. . . . . . . . . . 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))
96	94, 95	sylib 208	. . . . . . . . . 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))
97	78, 96	eqtr3d 2658	. . . . . . . . 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))
98	11	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → (𝐴 × {𝑅}):𝐴⟶Top)
99		inss2 3834	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ Fin
100	99, 46	sseldi 3601	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑘 ∈ Fin)
101		simplrr 801	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → 𝑣 ∈ 𝑅)
102	26	topopn 20711	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
103	102	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → ∪ 𝑅 ∈ 𝑅)
104	101, 103	ifcld 4131	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ∈ 𝑅)
105		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → 𝑅 ∈ Top)
106		fvconst2g 6467	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Top ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑅})‘𝑥) = 𝑅)
107	105, 106	sylan 488	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝑅})‘𝑥) = 𝑅)
108	104, 107	eleqtrrd 2704	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ∈ ((𝐴 × {𝑅})‘𝑥))
109		eldifn 3733	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝑘) → ¬ 𝑥 ∈ 𝑘)
110	109	iffalsed 4097	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝑘) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) = ∪ 𝑅)
111	110	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) = ∪ 𝑅)
112		eldifi 3732	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ 𝑘) → 𝑥 ∈ 𝐴)
113	112, 107	sylan2 491	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → ((𝐴 × {𝑅})‘𝑥) = 𝑅)
114	113	unieqd 4446	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → ∪ ((𝐴 × {𝑅})‘𝑥) = ∪ 𝑅)
115	111, 114	eqtr4d 2659	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ 𝑘)) → if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) = ∪ ((𝐴 × {𝑅})‘𝑥))
116	89, 98, 100, 108, 115	ptopn 21386	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → X𝑥 ∈ 𝐴 if(𝑥 ∈ 𝑘, 𝑣, ∪ 𝑅) ∈ (∏t‘(𝐴 × {𝑅})))
117	97, 116	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ (∏t‘(𝐴 × {𝑅})))
118		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (𝑥 ∈ (∏t‘(𝐴 × {𝑅})) ↔ {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ∈ (∏t‘(𝐴 × {𝑅}))))
119	117, 118	syl5ibrcom 237	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝑘 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑣 ∈ 𝑅)) → (𝑥 = {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → 𝑥 ∈ (∏t‘(𝐴 × {𝑅}))))
120	119	rexlimdvva 3038	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∃𝑘 ∈ (𝒫 𝐴 ∩ Fin)∃𝑣 ∈ 𝑅 𝑥 = {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → 𝑥 ∈ (∏t‘(𝐴 × {𝑅}))))
121	120	abssdv 3676	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → {𝑥 ∣ ∃𝑘 ∈ (𝒫 𝐴 ∩ Fin)∃𝑣 ∈ 𝑅 𝑥 = {𝑓 ∈ (∪ 𝑅 ↑𝑚 𝐴) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ⊆ (∏t‘(𝐴 × {𝑅})))
122	37, 121	eqsstrd 3639	. . . 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})))
123		tgfiss 20795	. . . 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})))
124	14, 122, 123	syl2anc 693	. . 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})))
125	9, 124	eqsstrd 3639	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑅 ^ko 𝒫 𝐴) ⊆ (∏t‘(𝐴 × {𝑅})))
126		eqid 2622	. . . . . . . 8 ⊢ (∏t‘(𝐴 × {𝑅})) = (∏t‘(𝐴 × {𝑅}))
127	126, 26	ptuniconst 21401	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → (∪ 𝑅 ↑𝑚 𝐴) = ∪ (∏t‘(𝐴 × {𝑅})))
128	127	ancoms 469	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∪ 𝑅 ↑𝑚 𝐴) = ∪ (∏t‘(𝐴 × {𝑅})))
129	30, 128	eqtrd 2656	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 Cn 𝑅) = ∪ (∏t‘(𝐴 × {𝑅})))
130	129	oveq2d 6666	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) = ((∏t‘(𝐴 × {𝑅})) ↾t ∪ (∏t‘(𝐴 × {𝑅}))))
131		eqid 2622	. . . . . 6 ⊢ ∪ (∏t‘(𝐴 × {𝑅})) = ∪ (∏t‘(𝐴 × {𝑅}))
132	131	restid 16094	. . . . 5 ⊢ ((∏t‘(𝐴 × {𝑅})) ∈ Top → ((∏t‘(𝐴 × {𝑅})) ↾t ∪ (∏t‘(𝐴 × {𝑅}))) = (∏t‘(𝐴 × {𝑅})))
133	14, 132	syl 17	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t ∪ (∏t‘(𝐴 × {𝑅}))) = (∏t‘(𝐴 × {𝑅})))
134	130, 133	eqtrd 2656	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) = (∏t‘(𝐴 × {𝑅})))
135	5, 126	xkoptsub 21457	. . . 4 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝑅 ∈ Top) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) ⊆ (𝑅 ^ko 𝒫 𝐴))
136	2, 3, 135	syl2anc 693	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → ((∏t‘(𝐴 × {𝑅})) ↾t (𝒫 𝐴 Cn 𝑅)) ⊆ (𝑅 ^ko 𝒫 𝐴))
137	134, 136	eqsstr3d 3640	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (∏t‘(𝐴 × {𝑅})) ⊆ (𝑅 ^ko 𝒫 𝐴))
138	125, 137	eqssd 3620	1 ⊢ ((𝑅 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑅 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   × cxp 5112  dom cdm 5114  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  ficfi 8316   ↾_t crest 16081  topGenctg 16098  ∏_tcpt 16099  Topctop 20698  TopOnctopon 20715   Cn ccn 21028  Compccmp 21189   ^_ko cxko 21364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-xko 21366

This theorem is referenced by:  tmdgsum  21899  tmdgsum2  21900  symgtgp  21905