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Theorem xkoinjcn 21538
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 = (𝑥𝑋 ↦ (𝑦𝑌 ↦ ⟨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.
StepHypRef Expression
1 simplr 807 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → 𝑆 ∈ (TopOn‘𝑌))
21cnmptid 21512 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → (𝑦𝑌𝑦) ∈ (𝑆 Cn 𝑆))
3 simpll 805 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → 𝑅 ∈ (TopOn‘𝑋))
4 simpr 476 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → 𝑥𝑋)
51, 3, 4cnmptc 21513 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → (𝑦𝑌𝑥) ∈ (𝑆 Cn 𝑅))
61, 2, 5cnmpt1t 21516 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×t 𝑅)))
7 xkoinjcn.3 . . 3 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))
86, 7fmptd 6425 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹:𝑋⟶(𝑆 Cn (𝑆 ×t 𝑅)))
9 eqid 2651 . . . . . 6 𝑆 = 𝑆
10 eqid 2651 . . . . . 6 {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp} = {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}
11 eqid 2651 . . . . . 6 (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣})
129, 10, 11xkobval 21437 . . . . 5 ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑧 ∣ ∃𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅)((𝑆t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣})}
1312abeq2i 2764 . . . 4 (𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅)((𝑆t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}))
14 simpll 805 . . . . . . . . . . . 12 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → (𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)))
1514, 6sylan 487 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×t 𝑅)))
16 imaeq1 5496 . . . . . . . . . . . . 13 (𝑓 = (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) → (𝑓𝑘) = ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘))
1716sseq1d 3665 . . . . . . . . . . . 12 (𝑓 = (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) → ((𝑓𝑘) ⊆ 𝑣 ↔ ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
1817elrab3 3397 . . . . . . . . . . 11 ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×t 𝑅)) → ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
1915, 18syl 17 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
20 funmpt 5964 . . . . . . . . . . 11 Fun (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)
21 simplrl 817 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑆)
2221elpwid 4203 . . . . . . . . . . . . . 14 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → 𝑘 𝑆)
2314simprd 478 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24 toponuni 20767 . . . . . . . . . . . . . . 15 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
2523, 24syl 17 . . . . . . . . . . . . . 14 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → 𝑌 = 𝑆)
2622, 25sseqtr4d 3675 . . . . . . . . . . . . 13 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → 𝑘𝑌)
2726adantr 480 . . . . . . . . . . . 12 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → 𝑘𝑌)
28 dmmptg 5670 . . . . . . . . . . . . 13 (∀𝑦𝑌𝑦, 𝑥⟩ ∈ V → dom (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌)
29 opex 4962 . . . . . . . . . . . . . 14 𝑦, 𝑥⟩ ∈ V
3029a1i 11 . . . . . . . . . . . . 13 (𝑦𝑌 → ⟨𝑦, 𝑥⟩ ∈ V)
3128, 30mprg 2955 . . . . . . . . . . . 12 dom (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌
3227, 31syl6sseqr 3685 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → 𝑘 ⊆ dom (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))
33 funimass4 6286 . . . . . . . . . . 11 ((Fun (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∧ 𝑘 ⊆ dom (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) → (((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣 ↔ ∀𝑧𝑘 ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣))
3420, 32, 33sylancr 696 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → (((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣 ↔ ∀𝑧𝑘 ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣))
3527sselda 3636 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) ∧ 𝑧𝑘) → 𝑧𝑌)
36 opeq1 4433 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ⟨𝑦, 𝑥⟩ = ⟨𝑧, 𝑥⟩)
37 eqid 2651 . . . . . . . . . . . . . . . 16 (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)
38 opex 4962 . . . . . . . . . . . . . . . 16 𝑧, 𝑥⟩ ∈ V
3936, 37, 38fvmpt 6321 . . . . . . . . . . . . . . 15 (𝑧𝑌 → ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) = ⟨𝑧, 𝑥⟩)
4035, 39syl 17 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) ∧ 𝑧𝑘) → ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) = ⟨𝑧, 𝑥⟩)
4140eleq1d 2715 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) ∧ 𝑧𝑘) → (((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
42 vex 3234 . . . . . . . . . . . . . 14 𝑥 ∈ V
43 opeq2 4434 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑥⟩)
4443eleq1d 2715 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
4542, 44ralsn 4254 . . . . . . . . . . . . 13 (∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣)
4641, 45syl6bbr 278 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) ∧ 𝑧𝑘) → (((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣))
4746ralbidva 3014 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → (∀𝑧𝑘 ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ∀𝑧𝑘𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣))
48 dfss3 3625 . . . . . . . . . . . 12 ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑡 ∈ (𝑘 × {𝑥})𝑡𝑣)
49 eleq1 2718 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑧, 𝑤⟩ → (𝑡𝑣 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑣))
5049ralxp 5296 . . . . . . . . . . . 12 (∀𝑡 ∈ (𝑘 × {𝑥})𝑡𝑣 ↔ ∀𝑧𝑘𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
5148, 50bitri 264 . . . . . . . . . . 11 ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑧𝑘𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
5247, 51syl6bbr 278 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → (∀𝑧𝑘 ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ (𝑘 × {𝑥}) ⊆ 𝑣))
5319, 34, 523bitrd 294 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑥𝑋) → ((𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ (𝑘 × {𝑥}) ⊆ 𝑣))
5453rabbidva 3219 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → {𝑥𝑋 ∣ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}} = {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})
55 sneq 4220 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → {𝑥} = {𝑤})
5655xpeq2d 5173 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑘 × {𝑥}) = (𝑘 × {𝑤}))
5756sseq1d 3665 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × {𝑤}) ⊆ 𝑣))
5857elrab 3396 . . . . . . . . . . 11 (𝑤 ∈ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣))
59 eqid 2651 . . . . . . . . . . . . 13 (𝑆t 𝑘) = (𝑆t 𝑘)
60 eqid 2651 . . . . . . . . . . . . 13 𝑅 = 𝑅
61 simplr 807 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆t 𝑘) ∈ Comp)
62 simpll 805 . . . . . . . . . . . . . . 15 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) → 𝑅 ∈ (TopOn‘𝑋))
6362ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑅 ∈ (TopOn‘𝑋))
64 topontop 20766 . . . . . . . . . . . . . 14 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
6563, 64syl 17 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑅 ∈ Top)
66 topontop 20766 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
6766adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ Top)
6864adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ Top)
69 txtop 21420 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ×t 𝑅) ∈ Top)
7067, 68, 69syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ×t 𝑅) ∈ Top)
7170ad3antrrr 766 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ×t 𝑅) ∈ Top)
72 vex 3234 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
73 toponmax 20778 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
7463, 73syl 17 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑋𝑅)
75 xpexg 7002 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ V ∧ 𝑋𝑅) → (𝑘 × 𝑋) ∈ V)
7672, 74, 75sylancr 696 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × 𝑋) ∈ V)
77 simprr 811 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) → 𝑣 ∈ (𝑆 ×t 𝑅))
7877ad2antrr 762 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑣 ∈ (𝑆 ×t 𝑅))
79 elrestr 16136 . . . . . . . . . . . . . . 15 (((𝑆 ×t 𝑅) ∈ Top ∧ (𝑘 × 𝑋) ∈ V ∧ 𝑣 ∈ (𝑆 ×t 𝑅)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ×t 𝑅) ↾t (𝑘 × 𝑋)))
8071, 76, 78, 79syl3anc 1366 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ×t 𝑅) ↾t (𝑘 × 𝑋)))
8167ad3antrrr 766 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑆 ∈ Top)
8272a1i 11 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 ∈ V)
83 txrest 21482 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ Top ∧ 𝑅 ∈ Top) ∧ (𝑘 ∈ V ∧ 𝑋𝑅)) → ((𝑆 ×t 𝑅) ↾t (𝑘 × 𝑋)) = ((𝑆t 𝑘) ×t (𝑅t 𝑋)))
8481, 65, 82, 74, 83syl22anc 1367 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ×t 𝑅) ↾t (𝑘 × 𝑋)) = ((𝑆t 𝑘) ×t (𝑅t 𝑋)))
85 toponuni 20767 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
8663, 85syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑋 = 𝑅)
8786oveq2d 6706 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅t 𝑋) = (𝑅t 𝑅))
8860restid 16141 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ (TopOn‘𝑋) → (𝑅t 𝑅) = 𝑅)
8963, 88syl 17 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅t 𝑅) = 𝑅)
9087, 89eqtrd 2685 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅t 𝑋) = 𝑅)
9190oveq2d 6706 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆t 𝑘) ×t (𝑅t 𝑋)) = ((𝑆t 𝑘) ×t 𝑅))
9284, 91eqtrd 2685 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ×t 𝑅) ↾t (𝑘 × 𝑋)) = ((𝑆t 𝑘) ×t 𝑅))
9380, 92eleqtrd 2732 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆t 𝑘) ×t 𝑅))
9423adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑆 ∈ (TopOn‘𝑌))
9526adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘𝑌)
96 resttopon 21013 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑘𝑌) → (𝑆t 𝑘) ∈ (TopOn‘𝑘))
9794, 95, 96syl2anc 694 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆t 𝑘) ∈ (TopOn‘𝑘))
98 toponuni 20767 . . . . . . . . . . . . . . . 16 ((𝑆t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = (𝑆t 𝑘))
9997, 98syl 17 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 = (𝑆t 𝑘))
10099xpeq1d 5172 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) = ( (𝑆t 𝑘) × {𝑤}))
101 simprr 811 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ 𝑣)
102 simprl 809 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑤𝑋)
103102snssd 4372 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → {𝑤} ⊆ 𝑋)
104 xpss2 5162 . . . . . . . . . . . . . . . 16 ({𝑤} ⊆ 𝑋 → (𝑘 × {𝑤}) ⊆ (𝑘 × 𝑋))
105103, 104syl 17 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ (𝑘 × 𝑋))
106101, 105ssind 3870 . . . . . . . . . . . . . 14 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
107100, 106eqsstr3d 3673 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ( (𝑆t 𝑘) × {𝑤}) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
108102, 86eleqtrd 2732 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑤 𝑅)
10959, 60, 61, 65, 93, 107, 108txtube 21491 . . . . . . . . . . . 12 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ∃𝑟𝑅 (𝑤𝑟 ∧ ( (𝑆t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
110 toponss 20779 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑟𝑅) → 𝑟𝑋)
11163, 110sylan 487 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → 𝑟𝑋)
112 ssrab 3713 . . . . . . . . . . . . . . . . 17 (𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑟𝑋 ∧ ∀𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
113112baib 964 . . . . . . . . . . . . . . . 16 (𝑟𝑋 → (𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ∀𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
114111, 113syl 17 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → (𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ∀𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
115 xpss2 5162 . . . . . . . . . . . . . . . . . 18 (𝑟𝑋 → (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))
116111, 115syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))
117116biantrud 527 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))))
118 iunid 4607 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑟 {𝑥} = 𝑟
119118xpeq2i 5170 . . . . . . . . . . . . . . . . . . 19 (𝑘 × 𝑥𝑟 {𝑥}) = (𝑘 × 𝑟)
120 xpiundi 5207 . . . . . . . . . . . . . . . . . . 19 (𝑘 × 𝑥𝑟 {𝑥}) = 𝑥𝑟 (𝑘 × {𝑥})
121119, 120eqtr3i 2675 . . . . . . . . . . . . . . . . . 18 (𝑘 × 𝑟) = 𝑥𝑟 (𝑘 × {𝑥})
122121sseq1i 3662 . . . . . . . . . . . . . . . . 17 ((𝑘 × 𝑟) ⊆ 𝑣 𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
123 iunss 4593 . . . . . . . . . . . . . . . . 17 ( 𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
124122, 123bitri 264 . . . . . . . . . . . . . . . 16 ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∀𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
125 ssin 3868 . . . . . . . . . . . . . . . 16 (((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋)) ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
126117, 124, 1253bitr3g 302 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → (∀𝑥𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
12799adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → 𝑘 = (𝑆t 𝑘))
128127xpeq1d 5172 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → (𝑘 × 𝑟) = ( (𝑆t 𝑘) × 𝑟))
129128sseq1d 3665 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → ((𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)) ↔ ( (𝑆t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
130114, 126, 1293bitrd 294 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → (𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ( (𝑆t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
131130anbi2d 740 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟𝑅) → ((𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) ↔ (𝑤𝑟 ∧ ( (𝑆t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))))
132131rexbidva 3078 . . . . . . . . . . . 12 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (∃𝑟𝑅 (𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) ↔ ∃𝑟𝑅 (𝑤𝑟 ∧ ( (𝑆t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))))
133109, 132mpbird 247 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ (𝑤𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ∃𝑟𝑅 (𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
13458, 133sylan2b 491 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) ∧ 𝑤 ∈ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) → ∃𝑟𝑅 (𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
135134ralrimiva 2995 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → ∀𝑤 ∈ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟𝑅 (𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
136 eltop2 20827 . . . . . . . . . 10 (𝑅 ∈ Top → ({𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅 ↔ ∀𝑤 ∈ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟𝑅 (𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})))
13714, 68, 1363syl 18 . . . . . . . . 9 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → ({𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅 ↔ ∀𝑤 ∈ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟𝑅 (𝑤𝑟𝑟 ⊆ {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})))
138135, 137mpbird 247 . . . . . . . 8 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → {𝑥𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅)
13954, 138eqeltrd 2730 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → {𝑥𝑋 ∣ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑅)
140 imaeq2 5497 . . . . . . . . 9 (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} → (𝐹𝑧) = (𝐹 “ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}))
1417mptpreima 5666 . . . . . . . . 9 (𝐹 “ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑥𝑋 ∣ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}}
142140, 141syl6eq 2701 . . . . . . . 8 (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} → (𝐹𝑧) = {𝑥𝑋 ∣ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}})
143142eleq1d 2715 . . . . . . 7 (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝐹𝑧) ∈ 𝑅 ↔ {𝑥𝑋 ∣ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑅))
144139, 143syl5ibrcom 237 . . . . . 6 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) ∧ (𝑆t 𝑘) ∈ Comp) → (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣} → (𝐹𝑧) ∈ 𝑅))
145144expimpd 628 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅))) → (((𝑆t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) → (𝐹𝑧) ∈ 𝑅))
146145rexlimdvva 3067 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (∃𝑘 ∈ 𝒫 𝑆𝑣 ∈ (𝑆 ×t 𝑅)((𝑆t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) → (𝐹𝑧) ∈ 𝑅))
14713, 146syl5bi 232 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) → (𝐹𝑧) ∈ 𝑅))
148147ralrimiv 2994 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣})(𝐹𝑧) ∈ 𝑅)
149 simpl 472 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
150 ovex 6718 . . . . . 6 (𝑆 Cn (𝑆 ×t 𝑅)) ∈ V
151150pwex 4878 . . . . 5 𝒫 (𝑆 Cn (𝑆 ×t 𝑅)) ∈ V
1529, 10, 11xkotf 21436 . . . . . 6 (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp} × (𝑆 ×t 𝑅))⟶𝒫 (𝑆 Cn (𝑆 ×t 𝑅))
153 frn 6091 . . . . . 6 ((𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp} × (𝑆 ×t 𝑅))⟶𝒫 (𝑆 Cn (𝑆 ×t 𝑅)) → ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑆 Cn (𝑆 ×t 𝑅)))
154152, 153ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑆 Cn (𝑆 ×t 𝑅))
155151, 154ssexi 4836 . . . 4 ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
156155a1i 11 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V)
1579, 10, 11xkoval 21438 . . . 4 ((𝑆 ∈ Top ∧ (𝑆 ×t 𝑅) ∈ Top) → ((𝑆 ×t 𝑅) ^ko 𝑆) = (topGen‘(fi‘ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}))))
15867, 70, 157syl2anc 694 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×t 𝑅) ^ko 𝑆) = (topGen‘(fi‘ran (𝑘 ∈ {𝑤 ∈ 𝒫 𝑆 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣}))))
159 eqid 2651 . . . . 5 ((𝑆 ×t 𝑅) ^ko 𝑆) = ((𝑆 ×t 𝑅) ^ko 𝑆)
160159xkotopon 21451 . . . 4 ((𝑆 ∈ Top ∧ (𝑆 ×t 𝑅) ∈ Top) → ((𝑆 ×t 𝑅) ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×t 𝑅))))
16167, 70, 160syl2anc 694 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×t 𝑅) ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×t 𝑅))))
162149, 156, 158, 161subbascn 21106 . 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 ∣ (𝑆t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×t 𝑅)) ∣ (𝑓𝑘) ⊆ 𝑣})(𝐹𝑧) ∈ 𝑅)))
1638, 148, 162mpbir2and 977 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   ciun 4552  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ficfi 8357  t crest 16128  topGenctg 16145  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  Compccmp 21237   ×t ctx 21411   ^ko cxko 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cnp 21080  df-cmp 21238  df-tx 21413  df-xko 21414
This theorem is referenced by:  cnmpt2k  21539
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