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Theorem xkoco2cn 21271
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 20880 . . . 4 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
51, 3, 4syl2anc 691 . . 3 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
6 eqid 2610 . . 3 (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔))
75, 6fmptd 6292 . 2 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
8 eqid 2610 . . . . . 6 𝑅 = 𝑅
9 eqid 2610 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
10 eqid 2610 . . . . . 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 21199 . . . . 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 2722 . . . 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 762 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
1513, 14, 4syl2anc 691 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
16 imaeq1 5380 . . . . . . . . . . . . . 14 ( = (𝐹𝑔) → (𝑘) = ((𝐹𝑔) “ 𝑘))
17 imaco 5557 . . . . . . . . . . . . . 14 ((𝐹𝑔) “ 𝑘) = (𝐹 “ (𝑔𝑘))
1816, 17syl6eq 2660 . . . . . . . . . . . . 13 ( = (𝐹𝑔) → (𝑘) = (𝐹 “ (𝑔𝑘)))
1918sseq1d 3595 . . . . . . . . . . . 12 ( = (𝐹𝑔) → ((𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
2019elrab3 3332 . . . . . . . . . . 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 2610 . . . . . . . . . . . . . . 15 𝑆 = 𝑆
23 eqid 2610 . . . . . . . . . . . . . . 15 𝑇 = 𝑇
2422, 23cnf 20860 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹: 𝑆 𝑇)
252, 24syl 17 . . . . . . . . . . . . 13 (𝜑𝐹: 𝑆 𝑇)
2625ad3antrrr 762 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹: 𝑆 𝑇)
27 ffun 5961 . . . . . . . . . . . 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 5396 . . . . . . . . . . . . 13 (𝑔𝑘) ⊆ ran 𝑔
308, 22cnf 20860 . . . . . . . . . . . . . . 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 5966 . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ 𝑆)
35 fdm 5964 . . . . . . . . . . . . 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 3605 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ dom 𝐹)
38 funimass3 6241 . . . . . . . . . . 11 ((Fun 𝐹 ∧ (𝑔𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3928, 37, 38syl2anc 691 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
4021, 39bitrd 267 . . . . . . . . 9 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
4140rabbidva 3163 . . . . . . . 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 758 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
44 cntop1 20854 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
452, 44syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
4645ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
47 simplrl 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
4847elpwid 4118 . . . . . . . . 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 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
51 simplrr 797 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
52 cnima 20879 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣𝑇) → (𝐹𝑣) ∈ 𝑆)
5350, 51, 52syl2anc 691 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑣) ∈ 𝑆)
548, 43, 46, 48, 49, 53xkoopn 21202 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔𝑘) ⊆ (𝐹𝑣)} ∈ (𝑆 ^ko 𝑅))
5541, 54eqeltrd 2688 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑆 ^ko 𝑅))
56 imaeq2 5381 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
576mptpreima 5545 . . . . . . . . 9 ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
5856, 57syl6eq 2660 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
5958eleq1d 2672 . . . . . . 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 236 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅)))
6160expimpd 627 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆 ^ko 𝑅)))
6261rexlimdvva 3020 . . . 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 231 . . 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 2948 . 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 2610 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
6665xkotopon 21213 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6742, 45, 66syl2anc 691 . . 3 (𝜑 → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
68 ovex 6577 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
6968pwex 4774 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
708, 9, 10xkotf 21198 . . . . . 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 5966 . . . . . 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 4731 . . . 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 20855 . . . . 5 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
762, 75syl 17 . . . 4 (𝜑𝑇 ∈ Top)
778, 9, 10xkoval 21200 . . . 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 691 . . 3 (𝜑 → (𝑇 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
79 eqid 2610 . . . . 5 (𝑇 ^ko 𝑅) = (𝑇 ^ko 𝑅)
8079xkotopon 21213 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
8142, 76, 80syl2anc 691 . . 3 (𝜑 → (𝑇 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
8267, 74, 78, 81subbascn 20868 . 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 959 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  wss 3540  𝒫 cpw 4108   cuni 4372  cmpt 4643   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  cima 5041  ccom 5042  Fun wfun 5798  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  ficfi 8199  t crest 15904  topGenctg 15921  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999   ^ko cxko 21174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cmp 21000  df-xko 21176
This theorem is referenced by:  cnmptk1  21294
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