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Theorem xkococn 21511
Description: Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
xkococn.1 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))
Assertion
Ref Expression
xkococn ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) Cn (𝑇 ^ko 𝑅)))
Distinct variable groups:   𝑓,𝑔,𝑅   𝑆,𝑓,𝑔   𝑇,𝑓,𝑔
Allowed substitution hints:   𝐹(𝑓,𝑔)

Proof of Theorem xkococn
Dummy variables 𝑘 𝑎 𝑣 𝑥 𝑦 𝑧 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 811 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 simprl 809 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑆 Cn 𝑇))
3 cnco 21118 . . . . 5 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝑓 ∈ (𝑆 Cn 𝑇)) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
41, 2, 3syl2anc 694 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
54ralrimivva 3000 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ∀𝑓 ∈ (𝑆 Cn 𝑇)∀𝑔 ∈ (𝑅 Cn 𝑆)(𝑓𝑔) ∈ (𝑅 Cn 𝑇))
6 xkococn.1 . . . 4 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))
76fmpt2 7282 . . 3 (∀𝑓 ∈ (𝑆 Cn 𝑇)∀𝑔 ∈ (𝑅 Cn 𝑆)(𝑓𝑔) ∈ (𝑅 Cn 𝑇) ↔ 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
85, 7sylib 208 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
9 eqid 2651 . . . . . . 7 (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})
109rnmpt2 6812 . . . . . 6 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑥 ∣ ∃𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
1110eleq2i 2722 . . . . 5 (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
12 abid 2639 . . . . 5 (𝑥 ∈ {𝑥 ∣ ∃𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ↔ ∃𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})
13 oveq2 6698 . . . . . . 7 (𝑦 = 𝑘 → (𝑅t 𝑦) = (𝑅t 𝑘))
1413eleq1d 2715 . . . . . 6 (𝑦 = 𝑘 → ((𝑅t 𝑦) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
1514rexrab 3403 . . . . 5 (∃𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑅((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
1611, 12, 153bitri 286 . . . 4 (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑅((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
178ad2antrr 762 . . . . . . . . . . . . 13 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
18 ffn 6083 . . . . . . . . . . . . 13 (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) → 𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
19 elpreima 6377 . . . . . . . . . . . . 13 (𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) → (𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ↔ (𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) ∧ (𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))
2017, 18, 193syl 18 . . . . . . . . . . . 12 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ↔ (𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) ∧ (𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))
21 coeq1 5312 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑎 → (𝑓𝑔) = (𝑎𝑔))
22 coeq2 5313 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑏 → (𝑎𝑔) = (𝑎𝑏))
23 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑎 ∈ V
24 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑏 ∈ V
2523, 24coex 7160 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑏) ∈ V
2621, 22, 6, 25ovmpt2 6838 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) → (𝑎𝐹𝑏) = (𝑎𝑏))
2726adantl 481 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → (𝑎𝐹𝑏) = (𝑎𝑏))
2827eleq1d 2715 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → ((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
29 imaeq1 5496 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑎𝑏) → (𝑘) = ((𝑎𝑏) “ 𝑘))
3029sseq1d 3665 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑎𝑏) → ((𝑘) ⊆ 𝑣 ↔ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3130elrab 3396 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ ((𝑎𝑏) ∈ (𝑅 Cn 𝑇) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3231simprbi 479 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)
33 simp2 1082 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ 𝑛-Locally Comp)
3433ad3antrrr 766 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑆 ∈ 𝑛-Locally Comp)
35 elpwi 4201 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ 𝒫 𝑅𝑘 𝑅)
3635ad2antrl 764 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → 𝑘 𝑅)
3736ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑘 𝑅)
38 simprr 811 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → (𝑅t 𝑘) ∈ Comp)
3938ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → (𝑅t 𝑘) ∈ Comp)
40 simplr 807 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑣𝑇)
41 simprll 819 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑎 ∈ (𝑆 Cn 𝑇))
42 simprlr 820 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → 𝑏 ∈ (𝑅 Cn 𝑆))
43 simprr 811 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)
446, 34, 37, 39, 40, 41, 42, 43xkococnlem 21510 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))
4544expr 642 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → (((𝑎𝑏) “ 𝑘) ⊆ 𝑣 → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
4632, 45syl5 34 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
4728, 46sylbid 230 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ (𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆))) → ((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
4847ralrimivva 3000 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ∀𝑎 ∈ (𝑆 Cn 𝑇)∀𝑏 ∈ (𝑅 Cn 𝑆)((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
49 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝐹‘⟨𝑎, 𝑏⟩))
50 df-ov 6693 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
5149, 50syl6eqr 2703 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝑎𝐹𝑏))
5251eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
53 eleq1 2718 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦𝑧 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑧))
5453anbi1d 741 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})) ↔ (⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5554rexbidv 3081 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑎, 𝑏⟩ → (∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})) ↔ ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5652, 55imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑎, 𝑏⟩ → (((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))) ↔ ((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))))
5756ralxp 5296 . . . . . . . . . . . . . . 15 (∀𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))) ↔ ∀𝑎 ∈ (𝑆 Cn 𝑇)∀𝑏 ∈ (𝑅 Cn 𝑆)((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5848, 57sylibr 224 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ∀𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5958r19.21bi 2961 . . . . . . . . . . . . 13 (((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) ∧ 𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))) → ((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
6059expimpd 628 . . . . . . . . . . . 12 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ((𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) ∧ (𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
6120, 60sylbid 230 . . . . . . . . . . 11 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
6261ralrimiv 2994 . . . . . . . . . 10 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ∀𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))
63 nllytop 21324 . . . . . . . . . . . . . . 15 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
64633ad2ant2 1103 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ Top)
65 simp3 1083 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑇 ∈ Top)
66 xkotop 21439 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
6764, 65, 66syl2anc 694 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
68 simp1 1081 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑅 ∈ Top)
69 xkotop 21439 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
7068, 64, 69syl2anc 694 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
71 txtop 21420 . . . . . . . . . . . . 13 (((𝑇 ^ko 𝑆) ∈ Top ∧ (𝑆 ^ko 𝑅) ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
7267, 70, 71syl2anc 694 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
7372ad2antrr 762 . . . . . . . . . . 11 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
74 eltop2 20827 . . . . . . . . . . 11 (((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top → ((𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ↔ ∀𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
7573, 74syl 17 . . . . . . . . . 10 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ((𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ↔ ∀𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
7662, 75mpbird 247 . . . . . . . . 9 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)))
77 imaeq2 5497 . . . . . . . . . 10 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) = (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
7877eleq1d 2715 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ↔ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
7976, 78syl5ibrcom 237 . . . . . . . 8 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
8079rexlimdva 3060 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → (∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
8180anassrs 681 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ 𝑘 ∈ 𝒫 𝑅) ∧ (𝑅t 𝑘) ∈ Comp) → (∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
8281expimpd 628 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ 𝑘 ∈ 𝒫 𝑅) → (((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → (𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
8382rexlimdva 3060 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (∃𝑘 ∈ 𝒫 𝑅((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → (𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
8416, 83syl5bi 232 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → (𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))))
8584ralrimiv 2994 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})(𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)))
86 eqid 2651 . . . . . 6 (𝑇 ^ko 𝑆) = (𝑇 ^ko 𝑆)
8786xkotopon 21451 . . . . 5 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
8864, 65, 87syl2anc 694 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
89 eqid 2651 . . . . . 6 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
9089xkotopon 21451 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9168, 64, 90syl2anc 694 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
92 txtopon 21442 . . . 4 (((𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)) ∧ (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆))) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ (TopOn‘((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))))
9388, 91, 92syl2anc 694 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ (TopOn‘((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))))
94 ovex 6718 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
9594pwex 4878 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
96 eqid 2651 . . . . . . 7 𝑅 = 𝑅
97 eqid 2651 . . . . . . 7 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
9896, 97, 9xkotf 21436 . . . . . 6 (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}):({𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} × 𝑇)⟶𝒫 (𝑅 Cn 𝑇)
99 frn 6091 . . . . . 6 ((𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}):({𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} × 𝑇)⟶𝒫 (𝑅 Cn 𝑇) → ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇))
10098, 99ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇)
10195, 100ssexi 4836 . . . 4 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V
102101a1i 11 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V)
10396, 97, 9xkoval 21438 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
1041033adant2 1100 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
105 eqid 2651 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
106105xkotopon 21451 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
1071063adant2 1100 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
10893, 102, 104, 107subbascn 21106 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝐹 ∈ (((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) Cn (𝑇 ^ko 𝑅)) ↔ (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) ∧ ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})(𝐹𝑥) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)))))
1098, 85, 108mpbir2and 977 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) Cn (𝑇 ^ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  cop 4216   cuni 4468   × cxp 5141  ccnv 5142  ran crn 5144  cima 5146  ccom 5147   Fn wfn 5921  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  𝑛-Locally cnlly 21316   ×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-ntr 20872  df-nei 20950  df-cn 21079  df-cmp 21238  df-nlly 21318  df-tx 21413  df-xko 21414
This theorem is referenced by:  cnmptkk  21534  xkofvcn  21535  symgtgp  21952
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