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Theorem xkoco1cn 21680
Description: If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 21681 independently of the more general xkococn 21683 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 21290 . . . 4 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
31, 2sylan 561 . . 3 ((𝜑𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
4 eqid 2770 . . 3 (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) = (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹))
53, 4fmptd 6527 . 2 (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)):(𝑆 Cn 𝑇)⟶(𝑅 Cn 𝑇))
6 eqid 2770 . . . . . 6 𝑅 = 𝑅
7 eqid 2770 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
8 eqid 2770 . . . . . 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})
96, 7, 8xkobval 21609 . . . . 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})}
109abeq2i 2883 . . . 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}))
111ad2antrr 697 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑅 Cn 𝑆))
1211, 2sylan 561 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → (𝑔𝐹) ∈ (𝑅 Cn 𝑇))
13 imaeq1 5602 . . . . . . . . . . . . 13 ( = (𝑔𝐹) → (𝑘) = ((𝑔𝐹) “ 𝑘))
14 imaco 5784 . . . . . . . . . . . . 13 ((𝑔𝐹) “ 𝑘) = (𝑔 “ (𝐹𝑘))
1513, 14syl6eq 2820 . . . . . . . . . . . 12 ( = (𝑔𝐹) → (𝑘) = (𝑔 “ (𝐹𝑘)))
1615sseq1d 3779 . . . . . . . . . . 11 ( = (𝑔𝐹) → ((𝑘) ⊆ 𝑣 ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1716elrab3 3514 . . . . . . . . . 10 ((𝑔𝐹) ∈ (𝑅 Cn 𝑇) → ((𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1812, 17syl 17 . . . . . . . . 9 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑆 Cn 𝑇)) → ((𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣))
1918rabbidva 3337 . . . . . . . 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})
20 eqid 2770 . . . . . . . . 9 𝑆 = 𝑆
21 cntop2 21265 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑆 ∈ Top)
221, 21syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
2322ad2antrr 697 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
24 xkoco1cn.t . . . . . . . . . 10 (𝜑𝑇 ∈ Top)
2524ad2antrr 697 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑇 ∈ Top)
26 imassrn 5618 . . . . . . . . . 10 (𝐹𝑘) ⊆ ran 𝐹
276, 20cnf 21270 . . . . . . . . . . 11 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝐹: 𝑅 𝑆)
28 frn 6193 . . . . . . . . . . 11 (𝐹: 𝑅 𝑆 → ran 𝐹 𝑆)
2911, 27, 283syl 18 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → ran 𝐹 𝑆)
3026, 29syl5ss 3761 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑘) ⊆ 𝑆)
31 imacmp 21420 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 Cn 𝑆) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
3211, 31sylancom 568 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑆t (𝐹𝑘)) ∈ Comp)
33 simplrr 755 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
3420, 23, 25, 30, 32, 33xkoopn 21612 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔 “ (𝐹𝑘)) ⊆ 𝑣} ∈ (𝑇 ^ko 𝑆))
3519, 34eqeltrd 2849 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑇 ^ko 𝑆))
36 imaeq2 5603 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) = ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
374mptpreima 5772 . . . . . . . . 9 ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
3836, 37syl6eq 2820 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) = {𝑔 ∈ (𝑆 Cn 𝑇) ∣ (𝑔𝐹) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
3938eleq1d 2834 . . . . . . 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)))
4035, 39syl5ibrcom 237 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
4140expimpd 441 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
4241rexlimdvva 3185 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) “ 𝑥) ∈ (𝑇 ^ko 𝑆)))
4310, 42syl5bi 232 . . 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 3113 . 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 2770 . . . . 5 (𝑇 ^ko 𝑆) = (𝑇 ^ko 𝑆)
4645xkotopon 21623 . . . 4 ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
4722, 24, 46syl2anc 565 . . 3 (𝜑 → (𝑇 ^ko 𝑆) ∈ (TopOn‘(𝑆 Cn 𝑇)))
48 ovex 6822 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
4948pwex 4976 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
506, 7, 8xkotf 21608 . . . . . 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 6193 . . . . . 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 4934 . . . 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 21264 . . . . 5 (𝐹 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top)
561, 55syl 17 . . . 4 (𝜑𝑅 ∈ Top)
576, 7, 8xkoval 21610 . . . 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, 24, 57syl2anc 565 . . 3 (𝜑 → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
59 eqid 2770 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
6059xkotopon 21623 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6156, 24, 60syl2anc 565 . . 3 (𝜑 → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
6247, 54, 58, 61subbascn 21278 . 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))))
635, 44, 62mpbir2and 684 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 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  wss 3721  𝒫 cpw 4295   cuni 4572  cmpt 4861   × cxp 5247  ccnv 5248  ran crn 5250  cima 5252  ccom 5253  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  ficfi 8471  t crest 16288  topGenctg 16305  Topctop 20917  TopOnctopon 20934   Cn ccn 21248  Compccmp 21409   ^ko cxko 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-fin 8112  df-fi 8472  df-rest 16290  df-topgen 16311  df-top 20918  df-topon 20935  df-bases 20970  df-cn 21251  df-cmp 21410  df-xko 21586
This theorem is referenced by:  cnmpt1k  21705
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