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Theorem xkoco1cn 23505
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 23506 independently of the more general xkococn 23508 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 23114 . . . 4 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) β†’ (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
31, 2sylan 579 . . 3 ((πœ‘ ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) β†’ (𝑔 ∘ 𝐹) ∈ (𝑅 Cn 𝑇))
43fmpttd 7107 . 2 (πœ‘ β†’ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)):(𝑆 Cn 𝑇)⟢(𝑅 Cn 𝑇))
5 eqid 2724 . . . . . 6 βˆͺ 𝑅 = βˆͺ 𝑅
6 eqid 2724 . . . . . 6 {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 βˆͺ 𝑅 ∣ (𝑅 β†Ύt 𝑦) ∈ Comp}
7 eqid 2724 . . . . . 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 23434 . . . . 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})}
98eqabri 2869 . . . 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 723 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝐹 ∈ (𝑅 Cn 𝑆))
1110, 2sylan 579 . . . . . . . . . 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 6045 . . . . . . . . . . . . 13 (β„Ž = (𝑔 ∘ 𝐹) β†’ (β„Ž β€œ π‘˜) = ((𝑔 ∘ 𝐹) β€œ π‘˜))
13 imaco 6241 . . . . . . . . . . . . 13 ((𝑔 ∘ 𝐹) β€œ π‘˜) = (𝑔 β€œ (𝐹 β€œ π‘˜))
1412, 13eqtrdi 2780 . . . . . . . . . . . 12 (β„Ž = (𝑔 ∘ 𝐹) β†’ (β„Ž β€œ π‘˜) = (𝑔 β€œ (𝐹 β€œ π‘˜)))
1514sseq1d 4006 . . . . . . . . . . 11 (β„Ž = (𝑔 ∘ 𝐹) β†’ ((β„Ž β€œ π‘˜) βŠ† 𝑣 ↔ (𝑔 β€œ (𝐹 β€œ π‘˜)) βŠ† 𝑣))
1615elrab3 3677 . . . . . . . . . 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 3431 . . . . . . . 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 2724 . . . . . . . . 9 βˆͺ 𝑆 = βˆͺ 𝑆
20 cntop2 23089 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) β†’ 𝑆 ∈ Top)
211, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑆 ∈ Top)
2221ad2antrr 723 . . . . . . . . 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 723 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝑇 ∈ Top)
25 imassrn 6061 . . . . . . . . . 10 (𝐹 β€œ π‘˜) βŠ† ran 𝐹
265, 19cnf 23094 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) β†’ 𝐹:βˆͺ π‘…βŸΆβˆͺ 𝑆)
27 frn 6715 . . . . . . . . . . 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 3986 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ (𝐹 β€œ π‘˜) βŠ† βˆͺ 𝑆)
30 imacmp 23245 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ (𝑆 β†Ύt (𝐹 β€œ π‘˜)) ∈ Comp)
3110, 30sylancom 587 . . . . . . . . 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 775 . . . . . . . . 9 (((πœ‘ ∧ (π‘˜ ∈ 𝒫 βˆͺ 𝑅 ∧ 𝑣 ∈ 𝑇)) ∧ (𝑅 β†Ύt π‘˜) ∈ Comp) β†’ 𝑣 ∈ 𝑇)
3319, 22, 24, 29, 31, 32xkoopn 23437 . . . . . . . 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 2825 . . . . . . 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 6046 . . . . . . . . 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 2724 . . . . . . . . . 10 (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹))
3736mptpreima 6228 . . . . . . . . 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 2780 . . . . . . . 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 2810 . . . . . . 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 246 . . . . . 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 453 . . . . 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 3203 . . . 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 3137 . 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 2724 . . . . 5 (𝑇 ↑ko 𝑆) = (𝑇 ↑ko 𝑆)
4645xkotopon 23448 . . . 4 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) β†’ (𝑇 ↑ko 𝑆) ∈ (TopOnβ€˜(𝑆 Cn 𝑇)))
4721, 23, 46syl2anc 583 . . 3 (πœ‘ β†’ (𝑇 ↑ko 𝑆) ∈ (TopOnβ€˜(𝑆 Cn 𝑇)))
48 ovex 7435 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
4948pwex 5369 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
505, 6, 7xkotf 23433 . . . . . 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 6715 . . . . . 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 5313 . . . 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 23088 . . . . 5 (𝐹 ∈ (𝑅 Cn 𝑆) β†’ 𝑅 ∈ Top)
561, 55syl 17 . . . 4 (πœ‘ β†’ 𝑅 ∈ Top)
575, 6, 7xkoval 23435 . . . 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 583 . . 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 2724 . . . . 5 (𝑇 ↑ko 𝑅) = (𝑇 ↑ko 𝑅)
6059xkotopon 23448 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) β†’ (𝑇 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑇)))
6156, 23, 60syl2anc 583 . . 3 (πœ‘ β†’ (𝑇 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑇)))
6247, 54, 58, 61subbascn 23102 . 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 710 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  {crab 3424  Vcvv 3466   βŠ† wss 3941  π’« cpw 4595  βˆͺ cuni 4900   ↦ cmpt 5222   Γ— cxp 5665  β—‘ccnv 5666  ran crn 5668   β€œ cima 5670   ∘ ccom 5671  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404  ficfi 9402   β†Ύt crest 17371  topGenctg 17388  Topctop 22739  TopOnctopon 22756   Cn ccn 23072  Compccmp 23234   ↑ko cxko 23409
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8462  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-fin 8940  df-fi 9403  df-rest 17373  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-cn 23075  df-cmp 23235  df-xko 23411
This theorem is referenced by:  cnmpt1k  23530
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