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Theorem xkococn 23608
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 ∈ (((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) Cn (𝑇ko 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 771 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 simprl 769 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑆 Cn 𝑇))
3 cnco 23214 . . . . 5 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝑓 ∈ (𝑆 Cn 𝑇)) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
41, 2, 3syl2anc 582 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
54ralrimivva 3190 . . 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))
76fmpo 8073 . . 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 217 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
9 eqid 2725 . . . . . . 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})
109rnmpo 7554 . . . . . 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 2817 . . . . 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 2706 . . . . 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 7427 . . . . . . 7 (𝑦 = 𝑘 → (𝑅t 𝑦) = (𝑅t 𝑘))
1413eleq1d 2810 . . . . . 6 (𝑦 = 𝑘 → ((𝑅t 𝑦) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
1514rexrab 3688 . . . . 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 296 . . . 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 724 . . . . . . . . . . . . 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 6723 . . . . . . . . . . . . 13 (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) → 𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
19 elpreima 7066 . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑎 → (𝑓𝑔) = (𝑎𝑔))
22 coeq2 5861 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑏 → (𝑎𝑔) = (𝑎𝑏))
23 vex 3465 . . . . . . . . . . . . . . . . . . . . 21 𝑎 ∈ V
24 vex 3465 . . . . . . . . . . . . . . . . . . . . 21 𝑏 ∈ V
2523, 24coex 7938 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑏) ∈ V
2621, 22, 6, 25ovmpo 7581 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) → (𝑎𝐹𝑏) = (𝑎𝑏))
2726adantl 480 . . . . . . . . . . . . . . . . . 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 2810 . . . . . . . . . . . . . . . . 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 6059 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑎𝑏) → (𝑘) = ((𝑎𝑏) “ 𝑘))
3029sseq1d 4008 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑎𝑏) → ((𝑘) ⊆ 𝑣 ↔ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3130elrab 3679 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ ((𝑎𝑏) ∈ (𝑅 Cn 𝑇) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3231simprbi 495 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)
33 simp2 1134 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ 𝑛-Locally Comp)
3433ad3antrrr 728 . . . . . . . . . . . . . . . . . . . 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 4611 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ 𝒫 𝑅𝑘 𝑅)
3635ad2antrl 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → 𝑘 𝑅)
3736ad2antrr 724 . . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → (𝑅t 𝑘) ∈ Comp)
3938ad2antrr 724 . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . 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 23607 . . . . . . . . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))
4544expr 455 . . . . . . . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
4728, 46sylbid 239 . . . . . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
4847ralrimivva 3190 . . . . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
49 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝐹‘⟨𝑎, 𝑏⟩))
50 df-ov 7422 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
5149, 50eqtr4di 2783 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝑎𝐹𝑏))
5251eleq1d 2810 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
53 eleq1 2813 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦𝑧 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑧))
5453anbi1d 629 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})) ↔ (⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5554rexbidv 3168 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑎, 𝑏⟩ → (∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})) ↔ ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5652, 55imbi12d 343 . . . . . . . . . . . . . . . 16 (𝑦 = ⟨𝑎, 𝑏⟩ → (((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))) ↔ ((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))))
5756ralxp 5844 . . . . . . . . . . . . . . 15 (∀𝑦 ∈ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))) ↔ ∀𝑎 ∈ (𝑆 Cn 𝑇)∀𝑏 ∈ (𝑅 Cn 𝑆)((𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5848, 57sylibr 233 . . . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5958r19.21bi 3238 . . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
6059expimpd 452 . . . . . . . . . . . 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 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
6120, 60sylbid 239 . . . . . . . . . . 11 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
6261ralrimiv 3134 . . . . . . . . . 10 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ∀𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})))
63 nllytop 23421 . . . . . . . . . . . . . . 15 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
64633ad2ant2 1131 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ Top)
65 simp3 1135 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑇 ∈ Top)
66 xkotop 23536 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ Top)
6764, 65, 66syl2anc 582 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ Top)
68 simp1 1133 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑅 ∈ Top)
69 xkotop 23536 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
7068, 64, 69syl2anc 582 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
71 txtop 23517 . . . . . . . . . . . . 13 (((𝑇ko 𝑆) ∈ Top ∧ (𝑆ko 𝑅) ∈ Top) → ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ∈ Top)
7267, 70, 71syl2anc 582 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ∈ Top)
7372ad2antrr 724 . . . . . . . . . . 11 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ∈ Top)
74 eltop2 22922 . . . . . . . . . . 11 (((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ∈ Top → ((𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ↔ ∀𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 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}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ↔ ∀𝑦 ∈ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
7662, 75mpbird 256 . . . . . . . . 9 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)))
77 imaeq2 6060 . . . . . . . . . 10 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) = (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
7877eleq1d 2810 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝐹𝑥) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ↔ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
7976, 78syl5ibrcom 246 . . . . . . . 8 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) ∧ 𝑣𝑇) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
8079rexlimdva 3144 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → (∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
8180anassrs 466 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ 𝑘 ∈ 𝒫 𝑅) ∧ (𝑅t 𝑘) ∈ Comp) → (∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
8281expimpd 452 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ 𝑘 ∈ 𝒫 𝑅) → (((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → (𝐹𝑥) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
8382rexlimdva 3144 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (∃𝑘 ∈ 𝒫 𝑅((𝑅t 𝑘) ∈ Comp ∧ ∃𝑣𝑇 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → (𝐹𝑥) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
8416, 83biimtrid 241 . . 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) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))))
8584ralrimiv 3134 . 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) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)))
86 eqid 2725 . . . . . 6 (𝑇ko 𝑆) = (𝑇ko 𝑆)
8786xkotopon 23548 . . . . 5 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
8864, 65, 87syl2anc 582 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
89 eqid 2725 . . . . . 6 (𝑆ko 𝑅) = (𝑆ko 𝑅)
9089xkotopon 23548 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9168, 64, 90syl2anc 582 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
92 txtopon 23539 . . . 4 (((𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)) ∧ (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆))) → ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ∈ (TopOn‘((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))))
9388, 91, 92syl2anc 582 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) ∈ (TopOn‘((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))))
94 ovex 7452 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
9594pwex 5380 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
96 eqid 2725 . . . . . . 7 𝑅 = 𝑅
97 eqid 2725 . . . . . . 7 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
9896, 97, 9xkotf 23533 . . . . . 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 6730 . . . . . 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 5323 . . . 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 23535 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
1041033adant2 1128 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
105 eqid 2725 . . . . 5 (𝑇ko 𝑅) = (𝑇ko 𝑅)
106105xkotopon 23548 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
1071063adant2 1128 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
10893, 102, 104, 107subbascn 23202 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝐹 ∈ (((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) Cn (𝑇ko 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) ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅)))))
1098, 85, 108mpbir2and 711 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) Cn (𝑇ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2702  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  wss 3944  𝒫 cpw 4604  cop 4636   cuni 4909   × cxp 5676  ccnv 5677  ran crn 5679  cima 5681  ccom 5682   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  ficfi 9435  t crest 17405  topGenctg 17422  Topctop 22839  TopOnctopon 22856   Cn ccn 23172  Compccmp 23334  𝑛-Locally cnlly 23413   ×t ctx 23508  ko cxko 23509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-top 22840  df-topon 22857  df-bases 22893  df-ntr 22968  df-nei 23046  df-cn 23175  df-cmp 23335  df-nlly 23415  df-tx 23510  df-xko 23511
This theorem is referenced by:  cnmptkk  23631  xkofvcn  23632  efmndtmd  24049
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