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Theorem xkoinjcn 21300

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 788	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (TopOn‘𝑌))
2	1	cnmptid 21274	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝑆 Cn 𝑆))
3		simpll 786	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ (TopOn‘𝑋))
4		simpr 476	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
5	1, 3, 4	cnmptc 21275	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝑥) ∈ (𝑆 Cn 𝑅))
6	1, 2, 5	cnmpt1t 21278	. . 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 6292	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹:𝑋⟶(𝑆 Cn (𝑆 ×_t 𝑅)))
9		eqid 2610	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
10		eqid 2610	. . . . . 6 ⊢ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} = {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}
11		eqid 2610	. . . . . 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 21199	. . . . 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 2722	. . . 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 786	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 5380	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) → (𝑓 “ 𝑘) = ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘))
17	16	sseq1d 3595	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
18	17	elrab3 3332	. . . . . . . . . . 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 5840	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
21		simplrl 796	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑆)
22	21	elpwid 4118	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ⊆ ∪ 𝑆)
23	14	simprd 478	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24		toponuni 20542	. . . . . . . . . . . . . . 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	sseqtr4d 3605	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ⊆ 𝑌)
27	26	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑘 ⊆ 𝑌)
28		dmmptg 5549	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ V → dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌)
29		opex 4859	. . . . . . . . . . . . . 14 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑌 → ⟨𝑦, 𝑥⟩ ∈ V)
31	28, 30	mprg 2910	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌
32	27, 31	syl6sseqr 3615	. . . . . . . . . . 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 6157	. . . . . . . . . . 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 694	. . . . . . . . . 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 3568	. . . . . . . . . . . . . . 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 4340	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ⟨𝑦, 𝑥⟩ = ⟨𝑧, 𝑥⟩)
37		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
38		opex 4859	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑧, 𝑥⟩ ∈ V
39	36, 37, 38	fvmpt 6191	. . . . . . . . . . . . . . 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 2672	. . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
43		opeq2 4341	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑥⟩)
44	43	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
45	42, 44	ralsn 4169	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣)
46	41, 45	syl6bbr 277	. . . . . . . . . . . 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 2968	. . . . . . . . . . 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 3558	. . . . . . . . . . . 12 ⊢ ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑡 ∈ (𝑘 × {𝑥})𝑡 ∈ 𝑣)
49		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (𝑡 ∈ 𝑣 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑣))
50	49	ralxp 5185	. . . . . . . . . . . 12 ⊢ (∀𝑡 ∈ (𝑘 × {𝑥})𝑡 ∈ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
51	48, 50	bitri 263	. . . . . . . . . . 11 ⊢ ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
52	47, 51	syl6bbr 277	. . . . . . . . . 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 293	. . . . . . . . 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 3163	. . . . . . . 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 4135	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤})
56	55	xpeq2d 5063	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑘 × {𝑥}) = (𝑘 × {𝑤}))
57	56	sseq1d 3595	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × {𝑤}) ⊆ 𝑣))
58	57	elrab 3331	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣))
59		eqid 2610	. . . . . . . . . . . . 13 ⊢ ∪ (𝑆 ↾_t 𝑘) = ∪ (𝑆 ↾_t 𝑘)
60		eqid 2610	. . . . . . . . . . . . 13 ⊢ ∪ 𝑅 = ∪ 𝑅
61		simplr 788	. . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → 𝑅 ∈ (TopOn‘𝑋))
63	62	ad2antrr 758	. . . . . . . . . . . . . 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 20541	. . . . . . . . . . . . . 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 20541	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
67	66	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ Top)
68	64	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ Top)
69		txtop 21182	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ×_t 𝑅) ∈ Top)
70	67, 68, 69	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ×_t 𝑅) ∈ Top)
71	70	ad3antrrr 762	. . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
73		toponmax 20543	. . . . . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ V ∧ 𝑋 ∈ 𝑅) → (𝑘 × 𝑋) ∈ V)
76	72, 74, 75	sylancr 694	. . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → 𝑣 ∈ (𝑆 ×_t 𝑅))
78	77	ad2antrr 758	. . . . . . . . . . . . . . 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 15912	. . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . 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 21244	. . . . . . . . . . . . . . . 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 1319	. . . . . . . . . . . . . . 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 20542	. . . . . . . . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . . . . 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 15917	. . . . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . 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 2690	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 20775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑘 ⊆ 𝑌) → (𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘))
97	94, 95, 96	syl2anc 691	. . . . . . . . . . . . . . . 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 20542	. . . . . . . . . . . . . . . 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 5062	. . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . 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 4281	. . . . . . . . . . . . . . . 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 5152	. . . . . . . . . . . . . . . 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 3799	. . . . . . . . . . . . . 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	eqsstr3d 3603	. . . . . . . . . . . . 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 2690	. . . . . . . . . . . . 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 21253	. . . . . . . . . . . 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 20544	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑟 ∈ 𝑅) → 𝑟 ⊆ 𝑋)
111	63, 110	sylan 487	. . . . . . . . . . . . . . . 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 3643	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑟 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
113	112	baib 942	. . . . . . . . . . . . . . . 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 5152	. . . . . . . . . . . . . . . . . 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 527	. . . . . . . . . . . . . . . 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 4511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑥 ∈ 𝑟 {𝑥} = 𝑟
119	118	xpeq2i 5060	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 × ∪ 𝑥 ∈ 𝑟 {𝑥}) = (𝑘 × 𝑟)
120		xpiundi 5096	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 × ∪ 𝑥 ∈ 𝑟 {𝑥}) = ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥})
121	119, 120	eqtr3i 2634	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 × 𝑟) = ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥})
122	121	sseq1i 3592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
123		iunss 4497	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
124	122, 123	bitri 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
125		ssin 3797	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋)) ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
126	117, 124, 125	3bitr3g 301	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 5062	. . . . . . . . . . . . . . . 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 3595	. . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . 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 736	. . . . . . . . . . . . 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 3031	. . . . . . . . . . . 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 246	. . . . . . . . . . 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 491	. . . . . . . . . 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 2949	. . . . . . . . 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 20590	. . . . . . . . . 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 246	. . . . . . . 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 2688	. . . . . . 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 5381	. . . . . . . . 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 5545	. . . . . . . . 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	syl6eq 2660	. . . . . . . 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 2672	. . . . . . 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 236	. . . . . 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 627	. . . . 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 3020	. . . 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 231	. . 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 2948	. 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 472	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
150		ovex 6577	. . . . . 6 ⊢ (𝑆 Cn (𝑆 ×_t 𝑅)) ∈ V
151	150	pwex 4774	. . . . 5 ⊢ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)) ∈ V
152	9, 10, 11	xkotf 21198	. . . . . 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 5966	. . . . . 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 4731	. . . 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 21200	. . . 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 691	. . 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 2610	. . . . 5 ⊢ ((𝑆 ×_t 𝑅) ^_ko 𝑆) = ((𝑆 ×_t 𝑅) ^_ko 𝑆)
160	159	xkotopon 21213	. . . 4 ⊢ ((𝑆 ∈ Top ∧ (𝑆 ×_t 𝑅) ∈ Top) → ((𝑆 ×_t 𝑅) ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×_t 𝑅))))
161	67, 70, 160	syl2anc 691	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×_t 𝑅) ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×_t 𝑅))))
162	149, 156, 158, 161	subbascn 20868	. 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 959	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ⟨cop 4131 ∪ cuni 4372 ∪ ciun 4455 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ficfi 8199 ↾_t crest 15904 topGenctg 15921 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 Compccmp 20999 ×_t ctx 21173 ^_ko cxko 21174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-fin 7845 df-fi 8200 df-rest 15906 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-cnp 20842 df-cmp 21000 df-tx 21175 df-xko 21176

This theorem is referenced by: cnmpt2k 21301

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