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Theorem xkoco1cn 22251
Description: If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 22252 independently of the more general xkococn 22254 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)
Hypotheses
Ref Expression
xkoco1cn.t (𝜑𝑇 ∈ Top)
xkoco1cn.f (𝜑𝐹 ∈ (𝑅 Cn 𝑆))
Assertion
Ref Expression
xkoco1cn (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇ko 𝑆) Cn (𝑇ko 𝑅)))
Distinct variable groups:   𝜑,𝑔   𝑅,𝑔   𝑆,𝑔   𝑇,𝑔   𝑔,𝐹

Proof of Theorem xkoco1cn
Dummy variables 𝑘 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkoco1cn.f . . . 4 (𝜑𝐹 ∈ (𝑅 Cn 𝑆))
2 cnco 21860 . . . 4 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
31, 2sylan 583 . . 3 ((𝜑𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
43fmpttd 6860 . 2 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇))
5 eqid 2824 . . . . . 6 𝑅 = 𝑅
6 eqid 2824 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
7 eqid 2824 . . . . . 6 (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})
85, 6, 7xkobval 22180 . . . . 5 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑥 ∣ ∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})}
98abeq2i 2951 . . . 4 (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
101ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑅 Cn 𝑆))
1110, 2sylan 583 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
12 imaeq1 5905 . . . . . . . . . . . . 13 ( = (𝑔𝐹) → (𝑘) = ((𝑔𝐹) “ 𝑘))
13 imaco 6085 . . . . . . . . . . . . 13 ((𝑔𝐹) “ 𝑘) = (𝑔 “ (𝐹𝑘))
1412, 13syl6eq 2875 . . . . . . . . . . . 12 ( = (𝑔𝐹) → (𝑘) = (𝑔 “ (𝐹𝑘)))
1514sseq1d 3982 . . . . . . . . . . 11 ( = (𝑔𝐹) → ((𝑘) ⊆ 𝑣 ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1615elrab3 3666 . . . . . . . . . 10 ((𝑔𝐹) ∈ (𝑅 Cn 𝑇) → ((𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1711, 16syl 17 . . . . . . . . 9 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → ((𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1817rabbidva 3463 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣})
19 eqid 2824 . . . . . . . . 9 𝑆 = 𝑆
20 cntop2 21835 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑆 ∈ Top)
211, 20syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
2221ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
23 xkoco1cn.t . . . . . . . . . 10 (𝜑𝑇 ∈ Top)
2423ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑇 ∈ Top)
25 imassrn 5921 . . . . . . . . . 10 (𝐹𝑘) ⊆ ran 𝐹
265, 19cnf 21840 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝐹: 𝑅 𝑆)
27 frn 6501 . . . . . . . . . . 11 (𝐹: 𝑅 𝑆 → ran 𝐹 𝑆)
2810, 26, 273syl 18 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → ran 𝐹 𝑆)
2925, 28sstrid 3962 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑘) ⊆ 𝑆)
30 imacmp 21991 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
3110, 30sylancom 591 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
32 simplrr 777 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
3319, 22, 24, 29, 31, 32xkoopn 22183 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣} ∈ (𝑇ko 𝑆))
3418, 33eqeltrd 2916 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑇ko 𝑆))
35 imaeq2 5906 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) = ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
36 eqid 2824 . . . . . . . . . 10 (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹))
3736mptpreima 6073 . . . . . . . . 9 ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
3835, 37syl6eq 2875 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
3938eleq1d 2900 . . . . . . 7 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆) ↔ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑇ko 𝑆)))
4034, 39syl5ibrcom 250 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
4140expimpd 457 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
4241rexlimdvva 3286 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
439, 42syl5bi 245 . . 3 (𝜑 → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
4443ralrimiv 3175 . 2 (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆))
45 eqid 2824 . . . . 5 (𝑇ko 𝑆) = (𝑇ko 𝑆)
4645xkotopon 22194 . . . 4 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
4721, 23, 46syl2anc 587 . . 3 (𝜑 → (𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
48 ovex 7171 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
4948pwex 5262 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
505, 6, 7xkotf 22179 . . . . . 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)
51 frn 6501 . . . . . 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))
5250, 51ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇)
5349, 52ssexi 5207 . . . 4 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V
5453a1i 11 . . 3 (𝜑 → ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V)
55 cntop1 21834 . . . . 5 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
561, 55syl 17 . . . 4 (𝜑𝑅 ∈ Top)
575, 6, 7xkoval 22181 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5856, 23, 57syl2anc 587 . . 3 (𝜑 → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
59 eqid 2824 . . . . 5 (𝑇ko 𝑅) = (𝑇ko 𝑅)
6059xkotopon 22194 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6156, 23, 60syl2anc 587 . . 3 (𝜑 → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6247, 54, 58, 61subbascn 21848 . 2 (𝜑 → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇ko 𝑆) Cn (𝑇ko 𝑅)) ↔ ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇) ∧ ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆))))
634, 44, 62mpbir2and 712 1 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇ko 𝑆) Cn (𝑇ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3132  wrex 3133  {crab 3136  Vcvv 3479  wss 3918  𝒫 cpw 4520   cuni 4819  cmpt 5127   × cxp 5534  ccnv 5535  ran crn 5537  cima 5539  ccom 5540  wf 6332  cfv 6336  (class class class)co 7138  cmpo 7140  ficfi 8858  t crest 16683  topGenctg 16700  Topctop 21487  TopOnctopon 21504   Cn ccn 21818  Compccmp 21980  ko cxko 22155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-iin 4903  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-fin 8496  df-fi 8859  df-rest 16685  df-topgen 16706  df-top 21488  df-topon 21505  df-bases 21540  df-cn 21821  df-cmp 21981  df-xko 22157
This theorem is referenced by:  cnmpt1k  22276
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