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Theorem xkoco1cn 23008
Description: If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 23009 independently of the more general xkococn 23011 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 22617 . . . 4 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
31, 2sylan 580 . . 3 ((𝜑𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
43fmpttd 7063 . 2 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇))
5 eqid 2736 . . . . . 6 𝑅 = 𝑅
6 eqid 2736 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
7 eqid 2736 . . . . . 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 22937 . . . . 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})}
98eqabi 2881 . . . 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 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑅 Cn 𝑆))
1110, 2sylan 580 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
12 imaeq1 6008 . . . . . . . . . . . . 13 ( = (𝑔𝐹) → (𝑘) = ((𝑔𝐹) “ 𝑘))
13 imaco 6203 . . . . . . . . . . . . 13 ((𝑔𝐹) “ 𝑘) = (𝑔 “ (𝐹𝑘))
1412, 13eqtrdi 2792 . . . . . . . . . . . 12 ( = (𝑔𝐹) → (𝑘) = (𝑔 “ (𝐹𝑘)))
1514sseq1d 3975 . . . . . . . . . . 11 ( = (𝑔𝐹) → ((𝑘) ⊆ 𝑣 ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1615elrab3 3646 . . . . . . . . . 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 3414 . . . . . . . 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 2736 . . . . . . . . 9 𝑆 = 𝑆
20 cntop2 22592 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑆 ∈ Top)
211, 20syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
2221ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
23 xkoco1cn.t . . . . . . . . . 10 (𝜑𝑇 ∈ Top)
2423ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑇 ∈ Top)
25 imassrn 6024 . . . . . . . . . 10 (𝐹𝑘) ⊆ ran 𝐹
265, 19cnf 22597 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝐹: 𝑅 𝑆)
27 frn 6675 . . . . . . . . . . 11 (𝐹: 𝑅 𝑆 → ran 𝐹 𝑆)
2810, 26, 273syl 18 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → ran 𝐹 𝑆)
2925, 28sstrid 3955 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑘) ⊆ 𝑆)
30 imacmp 22748 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
3110, 30sylancom 588 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
32 simplrr 776 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
3319, 22, 24, 29, 31, 32xkoopn 22940 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣} ∈ (𝑇ko 𝑆))
3418, 33eqeltrd 2838 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑇ko 𝑆))
35 imaeq2 6009 . . . . . . . . 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 2736 . . . . . . . . . 10 (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹))
3736mptpreima 6190 . . . . . . . . 9 ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
3835, 37eqtrdi 2792 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
3938eleq1d 2822 . . . . . . 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 246 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
4140expimpd 454 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
4241rexlimdvva 3205 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇ko 𝑆)))
439, 42biimtrid 241 . . 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 3142 . 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 2736 . . . . 5 (𝑇ko 𝑆) = (𝑇ko 𝑆)
4645xkotopon 22951 . . . 4 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
4721, 23, 46syl2anc 584 . . 3 (𝜑 → (𝑇ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
48 ovex 7390 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
4948pwex 5335 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
505, 6, 7xkotf 22936 . . . . . 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 6675 . . . . . 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 5279 . . . 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 22591 . . . . 5 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
561, 55syl 17 . . . 4 (𝜑𝑅 ∈ Top)
575, 6, 7xkoval 22938 . . . 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 584 . . 3 (𝜑 → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
59 eqid 2736 . . . . 5 (𝑇ko 𝑅) = (𝑇ko 𝑅)
6059xkotopon 22951 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6156, 23, 60syl2anc 584 . . 3 (𝜑 → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6247, 54, 58, 61subbascn 22605 . 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 711 1 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇ko 𝑆) Cn (𝑇ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560   cuni 4865  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  ficfi 9346  t crest 17302  topGenctg 17319  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Compccmp 22737  ko cxko 22912
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cmp 22738  df-xko 22914
This theorem is referenced by:  cnmpt1k  23033
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