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Theorem xkoco2cn 21509
Description: If 𝐹 is a continuous function, then 𝑔𝐹𝑔 is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypotheses
Ref Expression
xkoco2cn.r (𝜑𝑅 ∈ Top)
xkoco2cn.f (𝜑𝐹 ∈ (𝑆 Cn 𝑇))
Assertion
Ref Expression
xkoco2cn (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆 ^ko 𝑅) Cn (𝑇 ^ko 𝑅)))
Distinct variable groups:   𝜑,𝑔   𝑅,𝑔   𝑆,𝑔   𝑇,𝑔   𝑔,𝐹

Proof of Theorem xkoco2cn
Dummy variables 𝑘 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . 4 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 xkoco2cn.f . . . . 5 (𝜑𝐹 ∈ (𝑆 Cn 𝑇))
32adantr 480 . . . 4 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
4 cnco 21118 . . . 4 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
51, 3, 4syl2anc 694 . . 3 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
6 eqid 2651 . . 3 (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔))
75, 6fmptd 6425 . 2 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
8 eqid 2651 . . . . . 6 𝑅 = 𝑅
9 eqid 2651 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
10 eqid 2651 . . . . . 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})
118, 9, 10xkobval 21437 . . . . 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})}
1211abeq2i 2764 . . . 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}))
13 simpr 476 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
142ad3antrrr 766 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
1513, 14, 4syl2anc 694 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
16 imaeq1 5496 . . . . . . . . . . . . . 14 ( = (𝐹𝑔) → (𝑘) = ((𝐹𝑔) “ 𝑘))
17 imaco 5678 . . . . . . . . . . . . . 14 ((𝐹𝑔) “ 𝑘) = (𝐹 “ (𝑔𝑘))
1816, 17syl6eq 2701 . . . . . . . . . . . . 13 ( = (𝐹𝑔) → (𝑘) = (𝐹 “ (𝑔𝑘)))
1918sseq1d 3665 . . . . . . . . . . . 12 ( = (𝐹𝑔) → ((𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
2019elrab3 3397 . . . . . . . . . . 11 ((𝐹𝑔) ∈ (𝑅 Cn 𝑇) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
2115, 20syl 17 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
22 eqid 2651 . . . . . . . . . . . . . . 15 𝑆 = 𝑆
23 eqid 2651 . . . . . . . . . . . . . . 15 𝑇 = 𝑇
2422, 23cnf 21098 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹: 𝑆 𝑇)
252, 24syl 17 . . . . . . . . . . . . 13 (𝜑𝐹: 𝑆 𝑇)
2625ad3antrrr 766 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹: 𝑆 𝑇)
27 ffun 6086 . . . . . . . . . . . 12 (𝐹: 𝑆 𝑇 → Fun 𝐹)
2826, 27syl 17 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
29 imassrn 5512 . . . . . . . . . . . . 13 (𝑔𝑘) ⊆ ran 𝑔
308, 22cnf 21098 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝑅 Cn 𝑆) → 𝑔: 𝑅 𝑆)
3113, 30syl 17 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔: 𝑅 𝑆)
32 frn 6091 . . . . . . . . . . . . . 14 (𝑔: 𝑅 𝑆 → ran 𝑔 𝑆)
3331, 32syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 𝑆)
3429, 33syl5ss 3647 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ 𝑆)
35 fdm 6089 . . . . . . . . . . . . 13 (𝐹: 𝑆 𝑇 → dom 𝐹 = 𝑆)
3626, 35syl 17 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = 𝑆)
3734, 36sseqtr4d 3675 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ dom 𝐹)
38 funimass3 6373 . . . . . . . . . . 11 ((Fun 𝐹 ∧ (𝑔𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3928, 37, 38syl2anc 694 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
4021, 39bitrd 268 . . . . . . . . 9 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
4140rabbidva 3219 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔𝑘) ⊆ (𝐹𝑣)})
42 xkoco2cn.r . . . . . . . . . 10 (𝜑𝑅 ∈ Top)
4342ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
44 cntop1 21092 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
452, 44syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
4645ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
47 simplrl 817 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
4847elpwid 4203 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 𝑅)
49 simpr 476 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑅t 𝑘) ∈ Comp)
502ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
51 simplrr 818 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
52 cnima 21117 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣𝑇) → (𝐹𝑣) ∈ 𝑆)
5350, 51, 52syl2anc 694 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑣) ∈ 𝑆)
548, 43, 46, 48, 49, 53xkoopn 21440 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔𝑘) ⊆ (𝐹𝑣)} ∈ (𝑆 ^ko 𝑅))
5541, 54eqeltrd 2730 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑆 ^ko 𝑅))
56 imaeq2 5497 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
576mptpreima 5666 . . . . . . . . 9 ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
5856, 57syl6eq 2701 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
5958eleq1d 2715 . . . . . . 7 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅) ↔ {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑆 ^ko 𝑅)))
6055, 59syl5ibrcom 237 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅)))
6160expimpd 628 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅)))
6261rexlimdvva 3067 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅)))
6312, 62syl5bi 232 . . 3 (𝜑 → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅)))
6463ralrimiv 2994 . 2 (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅))
65 eqid 2651 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
6665xkotopon 21451 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6742, 45, 66syl2anc 694 . . 3 (𝜑 → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
68 ovex 6718 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
6968pwex 4878 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
708, 9, 10xkotf 21436 . . . . . 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)
71 frn 6091 . . . . . 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))
7270, 71ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇)
7369, 72ssexi 4836 . . . 4 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V
7473a1i 11 . . 3 (𝜑 → ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V)
75 cntop2 21093 . . . . 5 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
762, 75syl 17 . . . 4 (𝜑𝑇 ∈ Top)
778, 9, 10xkoval 21438 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
7842, 76, 77syl2anc 694 . . 3 (𝜑 → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
79 eqid 2651 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
8079xkotopon 21451 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
8142, 76, 80syl2anc 694 . . 3 (𝜑 → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
8267, 74, 78, 81subbascn 21106 . 2 (𝜑 → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆 ^ko 𝑅) Cn (𝑇 ^ko 𝑅)) ↔ ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇) ∧ ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅))))
837, 64, 82mpbir2and 977 1 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆 ^ko 𝑅) Cn (𝑇 ^ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ficfi 8357  t crest 16128  topGenctg 16145  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  Compccmp 21237   ^ko cxko 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cmp 21238  df-xko 21414
This theorem is referenced by:  cnmptk1  21532
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