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Theorem xkococn 21373
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 795 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 simprl 793 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑆 Cn 𝑇))
3 cnco 20980 . . . . 5 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝑓 ∈ (𝑆 Cn 𝑇)) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
41, 2, 3syl2anc 692 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
54ralrimivva 2965 . . 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 7182 . . 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 2621 . . . . . . 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 6723 . . . . . 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 2690 . . . . 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 2609 . . . . 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 6612 . . . . . . 7 (𝑦 = 𝑘 → (𝑅t 𝑦) = (𝑅t 𝑘))
1413eleq1d 2683 . . . . . 6 (𝑦 = 𝑘 → ((𝑅t 𝑦) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
1514rexrab 3352 . . . . 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 761 . . . . . . . . . . . . 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 6002 . . . . . . . . . . . . 13 (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) → 𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
19 elpreima 6293 . . . . . . . . . . . . 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 5239 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑎 → (𝑓𝑔) = (𝑎𝑔))
22 coeq2 5240 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑏 → (𝑎𝑔) = (𝑎𝑏))
23 vex 3189 . . . . . . . . . . . . . . . . . . . . 21 𝑎 ∈ V
24 vex 3189 . . . . . . . . . . . . . . . . . . . . 21 𝑏 ∈ V
2523, 24coex 7065 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑏) ∈ V
2621, 22, 6, 25ovmpt2 6749 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) → (𝑎𝐹𝑏) = (𝑎𝑏))
2726adantl 482 . . . . . . . . . . . . . . . . . 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 2683 . . . . . . . . . . . . . . . . 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 5420 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑎𝑏) → (𝑘) = ((𝑎𝑏) “ 𝑘))
3029sseq1d 3611 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑎𝑏) → ((𝑘) ⊆ 𝑣 ↔ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3130elrab 3346 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ ((𝑎𝑏) ∈ (𝑅 Cn 𝑇) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3231simprbi 480 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)
33 simp2 1060 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ 𝑛-Locally Comp)
3433ad3antrrr 765 . . . . . . . . . . . . . . . . . . . 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 4140 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ 𝒫 𝑅𝑘 𝑅)
3635ad2antrl 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → 𝑘 𝑅)
3736ad2antrr 761 . . . . . . . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → (𝑅t 𝑘) ∈ Comp)
3938ad2antrr 761 . . . . . . . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . . . . . 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 21372 . . . . . . . . . . . . . . . . . . 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 2965 . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝐹‘⟨𝑎, 𝑏⟩))
50 df-ov 6607 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
5149, 50syl6eqr 2673 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝑎𝐹𝑏))
5251eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
53 eleq1 2686 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦𝑧 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑧))
5453anbi1d 740 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})) ↔ (⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5554rexbidv 3045 . . . . . . . . . . . . . . . . 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 334 . . . . . . . . . . . . . . . 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 5223 . . . . . . . . . . . . . . 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 2927 . . . . . . . . . . . . 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 2959 . . . . . . . . . 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 21186 . . . . . . . . . . . . . . 15 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
64633ad2ant2 1081 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ Top)
65 simp3 1061 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑇 ∈ Top)
66 xkotop 21301 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
6764, 65, 66syl2anc 692 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
68 simp1 1059 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑅 ∈ Top)
69 xkotop 21301 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
7068, 64, 69syl2anc 692 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
71 txtop 21282 . . . . . . . . . . . . 13 (((𝑇 ^ko 𝑆) ∈ Top ∧ (𝑆 ^ko 𝑅) ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
7267, 70, 71syl2anc 692 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
7372ad2antrr 761 . . . . . . . . . . 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 20690 . . . . . . . . . . 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 5421 . . . . . . . . . 10 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) = (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
7877eleq1d 2683 . . . . . . . . 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 3024 . . . . . . 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 679 . . . . . 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 3024 . . . 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 2959 . 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 2621 . . . . . 6 (𝑇 ^ko 𝑆) = (𝑇 ^ko 𝑆)
8786xkotopon 21313 . . . . 5 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
8864, 65, 87syl2anc 692 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
89 eqid 2621 . . . . . 6 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
9089xkotopon 21313 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9168, 64, 90syl2anc 692 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
92 txtopon 21304 . . . 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 692 . . 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 6632 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
9594pwex 4808 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
96 eqid 2621 . . . . . . 7 𝑅 = 𝑅
97 eqid 2621 . . . . . . 7 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
9896, 97, 9xkotf 21298 . . . . . 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 6010 . . . . . 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 4763 . . . 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 21300 . . . 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 1078 . . 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 2621 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
106105xkotopon 21313 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
1071063adant2 1078 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
10893, 102, 104, 107subbascn 20968 . 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 956 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 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  cop 4154   cuni 4402   × cxp 5072  ccnv 5073  ran crn 5075  cima 5077  ccom 5078   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  ficfi 8260  t crest 16002  topGenctg 16019  Topctop 20617  TopOnctopon 20618   Cn ccn 20938  Compccmp 21099  𝑛-Locally cnlly 21178   ×t ctx 21273   ^ko cxko 21274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-ntr 20734  df-nei 20812  df-cn 20941  df-cmp 21100  df-nlly 21180  df-tx 21275  df-xko 21276
This theorem is referenced by:  cnmptkk  21396  xkofvcn  21397  symgtgp  21815
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