MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkococn Structured version   Visualization version   GIF version

Theorem xkococn 21677
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 780 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 simprl 778 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑆 Cn 𝑇))
3 cnco 21284 . . . . 5 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝑓 ∈ (𝑆 Cn 𝑇)) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
41, 2, 3syl2anc 575 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑓 ∈ (𝑆 Cn 𝑇) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓𝑔) ∈ (𝑅 Cn 𝑇))
54ralrimivva 3159 . . 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 7470 . . 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 209 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇))
9 eqid 2806 . . . . . . 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 7000 . . . . . 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 2877 . . . . 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 2794 . . . . 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 6882 . . . . . . 7 (𝑦 = 𝑘 → (𝑅t 𝑦) = (𝑅t 𝑘))
1413eleq1d 2870 . . . . . 6 (𝑦 = 𝑘 → ((𝑅t 𝑦) ∈ Comp ↔ (𝑅t 𝑘) ∈ Comp))
1514rexrab 3566 . . . . 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 288 . . . 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 708 . . . . . . . . . . . . 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 6256 . . . . . . . . . . . . 13 (𝐹:((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))⟶(𝑅 Cn 𝑇) → 𝐹 Fn ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)))
19 elpreima 6559 . . . . . . . . . . . . 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 5481 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑎 → (𝑓𝑔) = (𝑎𝑔))
22 coeq2 5482 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑏 → (𝑎𝑔) = (𝑎𝑏))
23 vex 3394 . . . . . . . . . . . . . . . . . . . . 21 𝑎 ∈ V
24 vex 3394 . . . . . . . . . . . . . . . . . . . . 21 𝑏 ∈ V
2523, 24coex 7348 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑏) ∈ V
2621, 22, 6, 25ovmpt2 7026 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ (𝑆 Cn 𝑇) ∧ 𝑏 ∈ (𝑅 Cn 𝑆)) → (𝑎𝐹𝑏) = (𝑎𝑏))
2726adantl 469 . . . . . . . . . . . . . . . . . 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 2870 . . . . . . . . . . . . . . . . 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 5671 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑎𝑏) → (𝑘) = ((𝑎𝑏) “ 𝑘))
3029sseq1d 3829 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑎𝑏) → ((𝑘) ⊆ 𝑣 ↔ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3130elrab 3559 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ ((𝑎𝑏) ∈ (𝑅 Cn 𝑇) ∧ ((𝑎𝑏) “ 𝑘) ⊆ 𝑣))
3231simprbi 486 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑎𝑏) “ 𝑘) ⊆ 𝑣)
33 simp2 1160 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ 𝑛-Locally Comp)
3433ad3antrrr 712 . . . . . . . . . . . . . . . . . . . 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 4361 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ 𝒫 𝑅𝑘 𝑅)
3635ad2antrl 710 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → 𝑘 𝑅)
3736ad2antrr 708 . . . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) ∧ (𝑘 ∈ 𝒫 𝑅 ∧ (𝑅t 𝑘) ∈ Comp)) → (𝑅t 𝑘) ∈ Comp)
3938ad2antrr 708 . . . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . . . 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 21676 . . . . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . 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 3159 . . . . . . . . . . . . . . 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 6408 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝐹‘⟨𝑎, 𝑏⟩))
50 df-ov 6877 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐹𝑏) = (𝐹‘⟨𝑎, 𝑏⟩)
5149, 50syl6eqr 2858 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝐹𝑦) = (𝑎𝐹𝑏))
5251eleq1d 2870 . . . . . . . . . . . . . . . . 17 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝐹𝑦) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑎𝐹𝑏) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
53 eleq1 2873 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦𝑧 ↔ ⟨𝑎, 𝑏⟩ ∈ 𝑧))
5453anbi1d 617 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})) ↔ (⟨𝑎, 𝑏⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5554rexbidv 3240 . . . . . . . . . . . . . . . . 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 335 . . . . . . . . . . . . . . . 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 5465 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . 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 3120 . . . . . . . . . . . . 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 443 . . . . . . . . . . . 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 231 . . . . . . . . . . 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 3153 . . . . . . . . . 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 21490 . . . . . . . . . . . . . . 15 (𝑆 ∈ 𝑛-Locally Comp → 𝑆 ∈ Top)
64633ad2ant2 1157 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑆 ∈ Top)
65 simp3 1161 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑇 ∈ Top)
66 xkotop 21605 . . . . . . . . . . . . . 14 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
6764, 65, 66syl2anc 575 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top)
68 simp1 1159 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝑅 ∈ Top)
69 xkotop 21605 . . . . . . . . . . . . . 14 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
7068, 64, 69syl2anc 575 . . . . . . . . . . . . 13 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
71 txtop 21586 . . . . . . . . . . . . 13 (((𝑇 ^ko 𝑆) ∈ Top ∧ (𝑆 ^ko 𝑅) ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
7267, 70, 71syl2anc 575 . . . . . . . . . . . 12 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∈ Top)
7372ad2antrr 708 . . . . . . . . . . 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 20993 . . . . . . . . . . 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 248 . . . . . . . . 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 5672 . . . . . . . . . 10 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (𝐹𝑥) = (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
7877eleq1d 2870 . . . . . . . . 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 238 . . . . . . . 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 3219 . . . . . . 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 455 . . . . . 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 443 . . . . 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 3219 . . . 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 233 . . 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 3153 . 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 2806 . . . . . 6 (𝑇 ^ko 𝑆) = (𝑇 ^ko 𝑆)
8786xkotopon 21617 . . . . 5 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
8864, 65, 87syl2anc 575 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
89 eqid 2806 . . . . . 6 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
9089xkotopon 21617 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9168, 64, 90syl2anc 575 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
92 txtopon 21608 . . . 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 575 . . 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 6906 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
9594pwex 5050 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
96 eqid 2806 . . . . . . 7 𝑅 = 𝑅
97 eqid 2806 . . . . . . 7 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
9896, 97, 9xkotf 21602 . . . . . 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 6262 . . . . . 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 4998 . . . 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 21604 . . . 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 1154 . . 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 2806 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
106105xkotopon 21617 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
1071063adant2 1154 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
10893, 102, 104, 107subbascn 21272 . 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 695 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  wss 3769  𝒫 cpw 4351  cop 4376   cuni 4630   × cxp 5309  ccnv 5310  ran crn 5312  cima 5314  ccom 5315   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6874  cmpt2 6876  ficfi 8555  t crest 16286  topGenctg 16303  Topctop 20911  TopOnctopon 20928   Cn ccn 21242  Compccmp 21403  𝑛-Locally cnlly 21482   ×t ctx 21577   ^ko cxko 21578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-fin 8196  df-fi 8556  df-rest 16288  df-topgen 16309  df-top 20912  df-topon 20929  df-bases 20964  df-ntr 21038  df-nei 21116  df-cn 21245  df-cmp 21404  df-nlly 21484  df-tx 21579  df-xko 21580
This theorem is referenced by:  cnmptkk  21700  xkofvcn  21701  symgtgp  22118
  Copyright terms: Public domain W3C validator