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Theorem xkoco1cn 21831
Description: If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 21832 independently of the more general xkococn 21834 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)) ∈ ((𝑇 ^ko 𝑆) Cn (𝑇 ^ko 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 21441 . . . 4 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
31, 2sylan 577 . . 3 ((𝜑𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
43fmpttd 6634 . 2 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇))
5 eqid 2825 . . . . . 6 𝑅 = 𝑅
6 eqid 2825 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
7 eqid 2825 . . . . . 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 21760 . . . . 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 2940 . . . 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 719 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑅 Cn 𝑆))
1110, 2sylan 577 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
12 imaeq1 5702 . . . . . . . . . . . . 13 ( = (𝑔𝐹) → (𝑘) = ((𝑔𝐹) “ 𝑘))
13 imaco 5881 . . . . . . . . . . . . 13 ((𝑔𝐹) “ 𝑘) = (𝑔 “ (𝐹𝑘))
1412, 13syl6eq 2877 . . . . . . . . . . . 12 ( = (𝑔𝐹) → (𝑘) = (𝑔 “ (𝐹𝑘)))
1514sseq1d 3857 . . . . . . . . . . 11 ( = (𝑔𝐹) → ((𝑘) ⊆ 𝑣 ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1615elrab3 3587 . . . . . . . . . 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 3401 . . . . . . . 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 2825 . . . . . . . . 9 𝑆 = 𝑆
20 cntop2 21416 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑆 ∈ Top)
211, 20syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
2221ad2antrr 719 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
23 xkoco1cn.t . . . . . . . . . 10 (𝜑𝑇 ∈ Top)
2423ad2antrr 719 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑇 ∈ Top)
25 imassrn 5718 . . . . . . . . . 10 (𝐹𝑘) ⊆ ran 𝐹
265, 19cnf 21421 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝐹: 𝑅 𝑆)
27 frn 6284 . . . . . . . . . . 11 (𝐹: 𝑅 𝑆 → ran 𝐹 𝑆)
2810, 26, 273syl 18 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → ran 𝐹 𝑆)
2925, 28syl5ss 3838 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑘) ⊆ 𝑆)
30 imacmp 21571 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
3110, 30sylancom 584 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
32 simplrr 798 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
3319, 22, 24, 29, 31, 32xkoopn 21763 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣} ∈ (𝑇 ^ko 𝑆))
3418, 33eqeltrd 2906 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑇 ^ko 𝑆))
35 imaeq2 5703 . . . . . . . . 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 2825 . . . . . . . . . 10 (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹))
3736mptpreima 5869 . . . . . . . . 9 ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
3835, 37syl6eq 2877 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
3938eleq1d 2891 . . . . . . 7 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆) ↔ {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑇 ^ko 𝑆)))
4034, 39syl5ibrcom 239 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
4140expimpd 447 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
4241rexlimdvva 3248 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
439, 42syl5bi 234 . . 3 (𝜑 → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
4443ralrimiv 3174 . 2 (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆))
45 eqid 2825 . . . . 5 (𝑇 ^ko 𝑆) = (𝑇 ^ko 𝑆)
4645xkotopon 21774 . . . 4 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
4721, 23, 46syl2anc 581 . . 3 (𝜑 → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
48 ovex 6937 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
4948pwex 5080 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
505, 6, 7xkotf 21759 . . . . . 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 6284 . . . . . 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 5028 . . . 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 21415 . . . . 5 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
561, 55syl 17 . . . 4 (𝜑𝑅 ∈ Top)
575, 6, 7xkoval 21761 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
5856, 23, 57syl2anc 581 . . 3 (𝜑 → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
59 eqid 2825 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
6059xkotopon 21774 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6156, 23, 60syl2anc 581 . . 3 (𝜑 → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6247, 54, 58, 61subbascn 21429 . 2 (𝜑 → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇 ^ko 𝑆) Cn (𝑇 ^ko 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) ∈ (𝑇 ^ko 𝑆))))
634, 44, 62mpbir2and 706 1 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇 ^ko 𝑆) Cn (𝑇 ^ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  wss 3798  𝒫 cpw 4378   cuni 4658  cmpt 4952   × cxp 5340  ccnv 5341  ran crn 5343  cima 5345  ccom 5346  wf 6119  cfv 6123  (class class class)co 6905  cmpt2 6907  ficfi 8585  t crest 16434  topGenctg 16451  Topctop 21068  TopOnctopon 21085   Cn ccn 21399  Compccmp 21560   ^ko cxko 21735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-fin 8226  df-fi 8586  df-rest 16436  df-topgen 16457  df-top 21069  df-topon 21086  df-bases 21121  df-cn 21402  df-cmp 21561  df-xko 21737
This theorem is referenced by:  cnmpt1k  21856
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