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Theorem xkoinjcn 22400

Description: Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)

**Hypothesis**
Ref	Expression
xkoinjcn.3	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))

**Assertion**
Ref	Expression
xkoinjcn	⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×_t 𝑅) ↑_ko 𝑆)))

Distinct variable groups: 𝑥,𝑦,𝑅 𝑥,𝑆,𝑦 𝑥,𝑌,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem xkoinjcn Dummy variables 𝑓 𝑘 𝑟 𝑣 𝑤 𝑧 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (TopOn‘𝑌))
2	1	cnmptid 22374	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝑆 Cn 𝑆))
3		simpll 766	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ (TopOn‘𝑋))
4		simpr 488	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
5	1, 3, 4	cnmptc 22375	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝑥) ∈ (𝑆 Cn 𝑅))
6	1, 2, 5	cnmpt1t 22378	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×_t 𝑅)))
7		xkoinjcn.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
8	6, 7	fmptd 6875	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹:𝑋⟶(𝑆 Cn (𝑆 ×_t 𝑅)))
9		eqid 2758	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
10		eqid 2758	. . . . . 6 ⊢ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} = {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}
11		eqid 2758	. . . . . 6 ⊢ (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
12	9, 10, 11	xkobval 22299	. . . . 5 ⊢ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑧 ∣ ∃𝑘 ∈ 𝒫 ∪ 𝑆∃𝑣 ∈ (𝑆 ×_t 𝑅)((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}
13	12	abeq2i 2887	. . . 4 ⊢ (𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 ∪ 𝑆∃𝑣 ∈ (𝑆 ×_t 𝑅)((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
14		simpll 766	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → (𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)))
15	14, 6	sylan 583	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×_t 𝑅)))
16		imaeq1 5901	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) → (𝑓 “ 𝑘) = ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘))
17	16	sseq1d 3925	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
18	17	elrab3 3605	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
19	15, 18	syl 17	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
20		funmpt 6378	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
21		simplrl 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑆)
22	21	elpwid 4508	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ⊆ ∪ 𝑆)
23	14	simprd 499	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24		toponuni 21627	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆)
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑌 = ∪ 𝑆)
26	22, 25	sseqtrrd 3935	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ⊆ 𝑌)
27	26	adantr 484	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑘 ⊆ 𝑌)
28		dmmptg 6076	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ V → dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌)
29		opex 5328	. . . . . . . . . . . . . 14 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑌 → ⟨𝑦, 𝑥⟩ ∈ V)
31	28, 30	mprg 3084	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌
32	27, 31	sseqtrrdi 3945	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑘 ⊆ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
33		funimass4 6723	. . . . . . . . . . 11 ⊢ ((Fun (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∧ 𝑘 ⊆ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣))
34	20, 32, 33	sylancr 590	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣))
35	27	sselda 3894	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → 𝑧 ∈ 𝑌)
36		opeq1 4764	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ⟨𝑦, 𝑥⟩ = ⟨𝑧, 𝑥⟩)
37		eqid 2758	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
38		opex 5328	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑧, 𝑥⟩ ∈ V
39	36, 37, 38	fvmpt 6764	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑌 → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) = ⟨𝑧, 𝑥⟩)
40	35, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) = ⟨𝑧, 𝑥⟩)
41	40	eleq1d 2836	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
42		vex 3413	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
43		opeq2 4766	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑥⟩)
44	43	eleq1d 2836	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
45	42, 44	ralsn 4579	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣)
46	41, 45	bitr4di 292	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣))
47	46	ralbidva 3125	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣))
48		dfss3 3882	. . . . . . . . . . . 12 ⊢ ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑡 ∈ (𝑘 × {𝑥})𝑡 ∈ 𝑣)
49		eleq1 2839	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (𝑡 ∈ 𝑣 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑣))
50	49	ralxp 5687	. . . . . . . . . . . 12 ⊢ (∀𝑡 ∈ (𝑘 × {𝑥})𝑡 ∈ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
51	48, 50	bitri 278	. . . . . . . . . . 11 ⊢ ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
52	47, 51	bitr4di 292	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ (𝑘 × {𝑥}) ⊆ 𝑣))
53	19, 34, 52	3bitrd 308	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ (𝑘 × {𝑥}) ⊆ 𝑣))
54	53	rabbidva 3390	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} = {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})
55		sneq 4535	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤})
56	55	xpeq2d 5558	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑘 × {𝑥}) = (𝑘 × {𝑤}))
57	56	sseq1d 3925	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × {𝑤}) ⊆ 𝑣))
58	57	elrab 3604	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣))
59		eqid 2758	. . . . . . . . . . . . 13 ⊢ ∪ (𝑆 ↾_t 𝑘) = ∪ (𝑆 ↾_t 𝑘)
60		eqid 2758	. . . . . . . . . . . . 13 ⊢ ∪ 𝑅 = ∪ 𝑅
61		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ↾_t 𝑘) ∈ Comp)
62		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → 𝑅 ∈ (TopOn‘𝑋))
63	62	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑅 ∈ (TopOn‘𝑋))
64		topontop 21626	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
65	63, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑅 ∈ Top)
66		topontop 21626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
67	66	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ Top)
68	64	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ Top)
69		txtop 22282	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ×_t 𝑅) ∈ Top)
70	67, 68, 69	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ×_t 𝑅) ∈ Top)
71	70	ad3antrrr 729	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ×_t 𝑅) ∈ Top)
72		vex 3413	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
73		toponmax 21639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅)
74	63, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑋 ∈ 𝑅)
75		xpexg 7477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ V ∧ 𝑋 ∈ 𝑅) → (𝑘 × 𝑋) ∈ V)
76	72, 74, 75	sylancr 590	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × 𝑋) ∈ V)
77		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → 𝑣 ∈ (𝑆 ×_t 𝑅))
78	77	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑣 ∈ (𝑆 ×_t 𝑅))
79		elrestr 16773	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ×_t 𝑅) ∈ Top ∧ (𝑘 × 𝑋) ∈ V ∧ 𝑣 ∈ (𝑆 ×_t 𝑅)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)))
80	71, 76, 78, 79	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)))
81	67	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑆 ∈ Top)
82	72	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 ∈ V)
83		txrest 22344	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ Top ∧ 𝑅 ∈ Top) ∧ (𝑘 ∈ V ∧ 𝑋 ∈ 𝑅)) → ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t (𝑅 ↾_t 𝑋)))
84	81, 65, 82, 74, 83	syl22anc 837	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t (𝑅 ↾_t 𝑋)))
85		toponuni 21627	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
86	63, 85	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑋 = ∪ 𝑅)
87	86	oveq2d 7172	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅 ↾_t 𝑋) = (𝑅 ↾_t ∪ 𝑅))
88	60	restid 16778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ (TopOn‘𝑋) → (𝑅 ↾_t ∪ 𝑅) = 𝑅)
89	63, 88	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅 ↾_t ∪ 𝑅) = 𝑅)
90	87, 89	eqtrd 2793	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅 ↾_t 𝑋) = 𝑅)
91	90	oveq2d 7172	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ↾_t 𝑘) ×_t (𝑅 ↾_t 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t 𝑅))
92	84, 91	eqtrd 2793	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t 𝑅))
93	80, 92	eleqtrd 2854	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ↾_t 𝑘) ×_t 𝑅))
94	23	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑆 ∈ (TopOn‘𝑌))
95	26	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 ⊆ 𝑌)
96		resttopon 21874	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑘 ⊆ 𝑌) → (𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘))
97	94, 95, 96	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘))
98		toponuni 21627	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = ∪ (𝑆 ↾_t 𝑘))
99	97, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 = ∪ (𝑆 ↾_t 𝑘))
100	99	xpeq1d 5557	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) = (∪ (𝑆 ↾_t 𝑘) × {𝑤}))
101		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ 𝑣)
102		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑤 ∈ 𝑋)
103	102	snssd 4702	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → {𝑤} ⊆ 𝑋)
104		xpss2 5548	. . . . . . . . . . . . . . . 16 ⊢ ({𝑤} ⊆ 𝑋 → (𝑘 × {𝑤}) ⊆ (𝑘 × 𝑋))
105	103, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ (𝑘 × 𝑋))
106	101, 105	ssind 4139	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
107	100, 106	eqsstrrd 3933	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (∪ (𝑆 ↾_t 𝑘) × {𝑤}) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
108	102, 86	eleqtrd 2854	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑤 ∈ ∪ 𝑅)
109	59, 60, 61, 65, 93, 107, 108	txtube 22353	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
110		toponss 21640	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑟 ∈ 𝑅) → 𝑟 ⊆ 𝑋)
111	63, 110	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ⊆ 𝑋)
112		ssrab 3979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑟 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
113	112	baib 539	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ⊆ 𝑋 → (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
114	111, 113	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
115		xpss2 5548	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ⊆ 𝑋 → (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))
116	111, 115	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))
117	116	biantrud 535	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))))
118		iunid 4952	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑥 ∈ 𝑟 {𝑥} = 𝑟
119	118	xpeq2i 5555	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 × ∪ 𝑥 ∈ 𝑟 {𝑥}) = (𝑘 × 𝑟)
120		xpiundi 5596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 × ∪ 𝑥 ∈ 𝑟 {𝑥}) = ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥})
121	119, 120	eqtr3i 2783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 × 𝑟) = ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥})
122	121	sseq1i 3922	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
123		iunss 4937	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
124	122, 123	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
125		ssin 4137	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋)) ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
126	117, 124, 125	3bitr3g 316	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
127	99	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → 𝑘 = ∪ (𝑆 ↾_t 𝑘))
128	127	xpeq1d 5557	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑘 × 𝑟) = (∪ (𝑆 ↾_t 𝑘) × 𝑟))
129	128	sseq1d 3925	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → ((𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)) ↔ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
130	114, 126, 129	3bitrd 308	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
131	130	anbi2d 631	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → ((𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) ↔ (𝑤 ∈ 𝑟 ∧ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))))
132	131	rexbidva 3220	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) ↔ ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))))
133	109, 132	mpbird 260	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
134	58, 133	sylan2b 596	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) → ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
135	134	ralrimiva 3113	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → ∀𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
136		eltop2 21688	. . . . . . . . . 10 ⊢ (𝑅 ∈ Top → ({𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅 ↔ ∀𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})))
137	14, 68, 136	3syl 18	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → ({𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅 ↔ ∀𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})))
138	135, 137	mpbird 260	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅)
139	54, 138	eqeltrd 2852	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑅)
140		imaeq2 5902	. . . . . . . . 9 ⊢ (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
141	7	mptpreima 6072	. . . . . . . . 9 ⊢ (^◡𝐹 “ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
142	140, 141	eqtrdi 2809	. . . . . . . 8 ⊢ (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑧) = {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}})
143	142	eleq1d 2836	. . . . . . 7 ⊢ (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → ((^◡𝐹 “ 𝑧) ∈ 𝑅 ↔ {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑅))
144	139, 143	syl5ibrcom 250	. . . . . 6 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑧) ∈ 𝑅))
145	144	expimpd 457	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → (((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑧) ∈ 𝑅))
146	145	rexlimdvva 3218	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (∃𝑘 ∈ 𝒫 ∪ 𝑆∃𝑣 ∈ (𝑆 ×_t 𝑅)((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑧) ∈ 𝑅))
147	13, 146	syl5bi 245	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑧) ∈ 𝑅))
148	147	ralrimiv 3112	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})(^◡𝐹 “ 𝑧) ∈ 𝑅)
149		simpl 486	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
150		ovex 7189	. . . . . 6 ⊢ (𝑆 Cn (𝑆 ×_t 𝑅)) ∈ V
151	150	pwex 5253	. . . . 5 ⊢ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)) ∈ V
152	9, 10, 11	xkotf 22298	. . . . . 6 ⊢ (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} × (𝑆 ×_t 𝑅))⟶𝒫 (𝑆 Cn (𝑆 ×_t 𝑅))
153		frn 6509	. . . . . 6 ⊢ ((𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} × (𝑆 ×_t 𝑅))⟶𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)) → ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)))
154	152, 153	ax-mp 5	. . . . 5 ⊢ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅))
155	151, 154	ssexi 5196	. . . 4 ⊢ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
156	155	a1i 11	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V)
157	9, 10, 11	xkoval 22300	. . . 4 ⊢ ((𝑆 ∈ Top ∧ (𝑆 ×_t 𝑅) ∈ Top) → ((𝑆 ×_t 𝑅) ↑_ko 𝑆) = (topGen‘(fi‘ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
158	67, 70, 157	syl2anc 587	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×_t 𝑅) ↑_ko 𝑆) = (topGen‘(fi‘ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
159		eqid 2758	. . . . 5 ⊢ ((𝑆 ×_t 𝑅) ↑_ko 𝑆) = ((𝑆 ×_t 𝑅) ↑_ko 𝑆)
160	159	xkotopon 22313	. . . 4 ⊢ ((𝑆 ∈ Top ∧ (𝑆 ×_t 𝑅) ∈ Top) → ((𝑆 ×_t 𝑅) ↑_ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×_t 𝑅))))
161	67, 70, 160	syl2anc 587	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×_t 𝑅) ↑_ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×_t 𝑅))))
162	149, 156, 158, 161	subbascn 21967	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑅 Cn ((𝑆 ×_t 𝑅) ↑_ko 𝑆)) ↔ (𝐹:𝑋⟶(𝑆 Cn (𝑆 ×_t 𝑅)) ∧ ∀𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})(^◡𝐹 “ 𝑧) ∈ 𝑅)))
163	8, 148, 162	mpbir2and 712	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×_t 𝑅) ↑_ko 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ∃wrex 3071 {crab 3074 Vcvv 3409 ∩ cin 3859 ⊆ wss 3860 𝒫 cpw 4497 {csn 4525 ⟨cop 4531 ∪ cuni 4801 ∪ ciun 4886 ↦ cmpt 5116 × cxp 5526 ^◡ccnv 5527 dom cdm 5528 ran crn 5529 “ cima 5531 Fun wfun 6334 ⟶wf 6336 ‘cfv 6340 (class class class)co 7156 ∈ cmpo 7158 ficfi 8920 ↾_t crest 16765 topGenctg 16782 Topctop 21606 TopOnctopon 21623 Cn ccn 21937 Compccmp 22099 ×_t ctx 22273 ↑_ko cxko 22274

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 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-1o 8118 df-er 8305 df-map 8424 df-en 8541 df-dom 8542 df-fin 8544 df-fi 8921 df-rest 16767 df-topgen 16788 df-top 21607 df-topon 21624 df-bases 21659 df-cn 21940 df-cnp 21941 df-cmp 22100 df-tx 22275 df-xko 22276

This theorem is referenced by: cnmpt2k 22401

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