MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkoco1cn Structured version   Visualization version   GIF version

Theorem xkoco1cn 23031
Description: If 𝐹 is a continuous function, then 𝑔 ↦ 𝑔 ∘ 𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 23032 independently of the more general xkococn 23034 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)
Hypotheses
Ref Expression
xkoco1cn.t (πœ‘ β†’ 𝑇 ∈ Top)
xkoco1cn.f (πœ‘ β†’ 𝐹 ∈ (𝑅 Cn 𝑆))
Assertion
Ref Expression
xkoco1cn (πœ‘ β†’ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))
Distinct variable groups:   πœ‘,𝑔   𝑅,𝑔   𝑆,𝑔   𝑇,𝑔   𝑔,𝐹

Proof of Theorem xkoco1cn
Dummy variables π‘˜ 𝑣 π‘₯ β„Ž 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkoco1cn.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑅 Cn 𝑆))
2 cnco 22640 . . . 4 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) β†’ (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
31, 2sylan 581 . . 3 ((πœ‘ ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) β†’ (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
43fmpttd 7067 . 2 (πœ‘ β†’ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)):(𝑆 Cn 𝑇)⟢(𝑅 Cn 𝑇))
5 eqid 2733 . . . . . 6 βˆͺ 𝑅 = βˆͺ 𝑅
6 eqid 2733 . . . . . 6 {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}
7 eqid 2733 . . . . . 6 (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) = (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣})
85, 6, 7xkobval 22960 . . . . 5 ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) = {π‘₯ ∣ βˆƒπ‘˜ ∈ 𝒫 βˆͺ π‘…βˆƒπ‘£ ∈ 𝑇 ((𝑅 β†Ύt π‘˜) ∈ Comp ∧ π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣})}
98eqabi 2878 . . . 4 (π‘₯ ∈ ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) ↔ βˆƒπ‘˜ ∈ 𝒫 βˆͺ π‘…βˆƒπ‘£ ∈ 𝑇 ((𝑅 β†Ύt π‘˜) ∈ Comp ∧ π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}))
101ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝐹 ∈ (𝑅 Cn 𝑆))
1110, 2sylan 581 . . . . . . . . . 10 ((((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) β†’ (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
12 imaeq1 6012 . . . . . . . . . . . . 13 (β„Ž = (𝑔 ∘ 𝐹) β†’ (β„Ž β€œ π‘˜) = ((𝑔 ∘ 𝐹) β€œ π‘˜))
13 imaco 6207 . . . . . . . . . . . . 13 ((𝑔 ∘ 𝐹) β€œ π‘˜) = (𝑔 β€œ (𝐹 β€œ π‘˜))
1412, 13eqtrdi 2789 . . . . . . . . . . . 12 (β„Ž = (𝑔 ∘ 𝐹) β†’ (β„Ž β€œ π‘˜) = (𝑔 β€œ (𝐹 β€œ π‘˜)))
1514sseq1d 3979 . . . . . . . . . . 11 (β„Ž = (𝑔 ∘ 𝐹) β†’ ((β„Ž β€œ π‘˜) βŠ† 𝑣 ↔ (𝑔 β€œ (𝐹 β€œ π‘˜)) βŠ† 𝑣))
1615elrab3 3650 . . . . . . . . . 10 ((𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇) β†’ ((𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣} ↔ (𝑔 β€œ (𝐹 β€œ π‘˜)) βŠ† 𝑣))
1711, 16syl 17 . . . . . . . . 9 ((((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) β†’ ((𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣} ↔ (𝑔 β€œ (𝐹 β€œ π‘˜)) βŠ† 𝑣))
1817rabbidva 3413 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}} = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 β€œ (𝐹 β€œ π‘˜)) βŠ† 𝑣})
19 eqid 2733 . . . . . . . . 9 βˆͺ 𝑆 = βˆͺ 𝑆
20 cntop2 22615 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) β†’ 𝑆 ∈ Top)
211, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑆 ∈ Top)
2221ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝑆 ∈ Top)
23 xkoco1cn.t . . . . . . . . . 10 (πœ‘ β†’ 𝑇 ∈ Top)
2423ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝑇 ∈ Top)
25 imassrn 6028 . . . . . . . . . 10 (𝐹 β€œ π‘˜) βŠ† ran 𝐹
265, 19cnf 22620 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) β†’ 𝐹:βˆͺ π‘…βŸΆβˆͺ 𝑆)
27 frn 6679 . . . . . . . . . . 11 (𝐹:βˆͺ π‘…βŸΆβˆͺ 𝑆 β†’ ran 𝐹 βŠ† βˆͺ 𝑆)
2810, 26, 273syl 18 . . . . . . . . . 10 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ ran 𝐹 βŠ† βˆͺ 𝑆)
2925, 28sstrid 3959 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ (𝐹 β€œ π‘˜) βŠ† βˆͺ 𝑆)
30 imacmp 22771 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ (𝑆 β†Ύt (𝐹 β€œ π‘˜)) ∈ Comp)
3110, 30sylancom 589 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ (𝑆 β†Ύt (𝐹 β€œ π‘˜)) ∈ Comp)
32 simplrr 777 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝑣 ∈ 𝑇)
3319, 22, 24, 29, 31, 32xkoopn 22963 . . . . . . . 8 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 β€œ (𝐹 β€œ π‘˜)) βŠ† 𝑣} ∈ (𝑇 ↑ko 𝑆))
3418, 33eqeltrd 2834 . . . . . . 7 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}} ∈ (𝑇 ↑ko 𝑆))
35 imaeq2 6013 . . . . . . . . 9 (π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣} β†’ (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) = (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}))
36 eqid 2733 . . . . . . . . . 10 (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹))
3736mptpreima 6194 . . . . . . . . 9 (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}}
3835, 37eqtrdi 2789 . . . . . . . 8 (π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣} β†’ (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}})
3938eleq1d 2819 . . . . . . 7 (π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣} β†’ ((β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆) ↔ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 ∘ 𝐹) ∈ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}} ∈ (𝑇 ↑ko 𝑆)))
4034, 39syl5ibrcom 247 . . . . . 6 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ (π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣} β†’ (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆)))
4140expimpd 455 . . . . 5 ((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) β†’ (((𝑅 β†Ύt π‘˜) ∈ Comp ∧ π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) β†’ (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆)))
4241rexlimdvva 3202 . . . 4 (πœ‘ β†’ (βˆƒπ‘˜ ∈ 𝒫 βˆͺ π‘…βˆƒπ‘£ ∈ 𝑇 ((𝑅 β†Ύt π‘˜) ∈ Comp ∧ π‘₯ = {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) β†’ (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆)))
439, 42biimtrid 241 . . 3 (πœ‘ β†’ (π‘₯ ∈ ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) β†’ (β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆)))
4443ralrimiv 3139 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣})(β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆))
45 eqid 2733 . . . . 5 (𝑇 ↑ko 𝑆) = (𝑇 ↑ko 𝑆)
4645xkotopon 22974 . . . 4 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) β†’ (𝑇 ↑ko 𝑆) ∈ (TopOnβ€˜(𝑆 Cn 𝑇)))
4721, 23, 46syl2anc 585 . . 3 (πœ‘ β†’ (𝑇 ↑ko 𝑆) ∈ (TopOnβ€˜(𝑆 Cn 𝑇)))
48 ovex 7394 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
4948pwex 5339 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
505, 6, 7xkotf 22959 . . . . . 6 (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}):({𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp} Γ— 𝑇)βŸΆπ’« (𝑅 Cn 𝑇)
51 frn 6679 . . . . . 6 ((π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}):({𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp} Γ— 𝑇)βŸΆπ’« (𝑅 Cn 𝑇) β†’ ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) βŠ† 𝒫 (𝑅 Cn 𝑇))
5250, 51ax-mp 5 . . . . 5 ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) βŠ† 𝒫 (𝑅 Cn 𝑇)
5349, 52ssexi 5283 . . . 4 ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) ∈ V
5453a1i 11 . . 3 (πœ‘ β†’ ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}) ∈ V)
55 cntop1 22614 . . . . 5 (𝐹 ∈ (𝑅 Cn 𝑆) β†’ 𝑅 ∈ Top)
561, 55syl 17 . . . 4 (πœ‘ β†’ 𝑅 ∈ Top)
575, 6, 7xkoval 22961 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) β†’ (𝑇 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}))))
5856, 23, 57syl2anc 585 . . 3 (πœ‘ β†’ (𝑇 ↑ko 𝑅) = (topGenβ€˜(fiβ€˜ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣}))))
59 eqid 2733 . . . . 5 (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
6059xkotopon 22974 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) β†’ (𝑇 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑇)))
6156, 23, 60syl2anc 585 . . 3 (πœ‘ β†’ (𝑇 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑇)))
6247, 54, 58, 61subbascn 22628 . 2 (πœ‘ β†’ ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)) ↔ ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)):(𝑆 Cn 𝑇)⟢(𝑅 Cn 𝑇) ∧ βˆ€π‘₯ ∈ ran (π‘˜ ∈ {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}, 𝑣 ∈ 𝑇 ↦ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ π‘˜) βŠ† 𝑣})(β—‘(𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) β€œ π‘₯) ∈ (𝑇 ↑ko 𝑆))))
634, 44, 62mpbir2and 712 1 (πœ‘ β†’ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869   ↦ cmpt 5192   Γ— cxp 5635  β—‘ccnv 5636  ran crn 5638   β€œ cima 5640   ∘ ccom 5641  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  ficfi 9354   β†Ύt crest 17310  topGenctg 17327  Topctop 22265  TopOnctopon 22282   Cn ccn 22598  Compccmp 22760   ↑ko cxko 22935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-cmp 22761  df-xko 22937
This theorem is referenced by:  cnmpt1k  23056
  Copyright terms: Public domain W3C validator