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Theorem xkoco2cn 22269
 Description: If 𝐹 is a continuous function, then 𝑔 ↦ 𝐹 ∘ 𝑔 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)) ∈ ((𝑆ko 𝑅) Cn (𝑇ko 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 488 . . . 4 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 xkoco2cn.f . . . . 5 (𝜑𝐹 ∈ (𝑆 Cn 𝑇))
32adantr 484 . . . 4 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
4 cnco 21877 . . . 4 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
51, 3, 4syl2anc 587 . . 3 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
65fmpttd 6870 . 2 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
7 eqid 2824 . . . . . 6 𝑅 = 𝑅
8 eqid 2824 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
9 eqid 2824 . . . . . 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})
107, 8, 9xkobval 22197 . . . . 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})}
1110abeq2i 2951 . . . 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}))
12 simpr 488 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
132ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
1412, 13, 4syl2anc 587 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
15 imaeq1 5911 . . . . . . . . . . . . . 14 ( = (𝐹𝑔) → (𝑘) = ((𝐹𝑔) “ 𝑘))
16 imaco 6091 . . . . . . . . . . . . . 14 ((𝐹𝑔) “ 𝑘) = (𝐹 “ (𝑔𝑘))
1715, 16syl6eq 2875 . . . . . . . . . . . . 13 ( = (𝐹𝑔) → (𝑘) = (𝐹 “ (𝑔𝑘)))
1817sseq1d 3984 . . . . . . . . . . . 12 ( = (𝐹𝑔) → ((𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
1918elrab3 3667 . . . . . . . . . . 11 ((𝐹𝑔) ∈ (𝑅 Cn 𝑇) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
2014, 19syl 17 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
21 eqid 2824 . . . . . . . . . . . . . . 15 𝑆 = 𝑆
22 eqid 2824 . . . . . . . . . . . . . . 15 𝑇 = 𝑇
2321, 22cnf 21857 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹: 𝑆 𝑇)
242, 23syl 17 . . . . . . . . . . . . 13 (𝜑𝐹: 𝑆 𝑇)
2524ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹: 𝑆 𝑇)
2625ffund 6507 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
27 imassrn 5927 . . . . . . . . . . . . 13 (𝑔𝑘) ⊆ ran 𝑔
287, 21cnf 21857 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝑅 Cn 𝑆) → 𝑔: 𝑅 𝑆)
2912, 28syl 17 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔: 𝑅 𝑆)
3029frnd 6510 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 𝑆)
3127, 30sstrid 3964 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ 𝑆)
3225fdmd 6513 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = 𝑆)
3331, 32sseqtrrd 3994 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ dom 𝐹)
34 funimass3 6815 . . . . . . . . . . 11 ((Fun 𝐹 ∧ (𝑔𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3526, 33, 34syl2anc 587 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3620, 35bitrd 282 . . . . . . . . 9 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3736rabbidva 3463 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔𝑘) ⊆ (𝐹𝑣)})
38 xkoco2cn.r . . . . . . . . . 10 (𝜑𝑅 ∈ Top)
3938ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
40 cntop1 21851 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
412, 40syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
4241ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
43 simplrl 776 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
4443elpwid 4533 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 𝑅)
45 simpr 488 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑅t 𝑘) ∈ Comp)
462ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
47 simplrr 777 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
48 cnima 21876 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣𝑇) → (𝐹𝑣) ∈ 𝑆)
4946, 47, 48syl2anc 587 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑣) ∈ 𝑆)
507, 39, 42, 44, 45, 49xkoopn 22200 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔𝑘) ⊆ (𝐹𝑣)} ∈ (𝑆ko 𝑅))
5137, 50eqeltrd 2916 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑆ko 𝑅))
52 imaeq2 5912 . . . . . . . . 9 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))
53 eqid 2824 . . . . . . . . . 10 (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔))
5453mptpreima 6079 . . . . . . . . 9 ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
5552, 54syl6eq 2875 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
5655eleq1d 2900 . . . . . . 7 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → (((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅) ↔ {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑆ko 𝑅)))
5751, 56syl5ibrcom 250 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
5857expimpd 457 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
5958rexlimdvva 3286 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
6011, 59syl5bi 245 . . 3 (𝜑 → (𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
6160ralrimiv 3176 . 2 (𝜑 → ∀𝑥 ∈ ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣})((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅))
62 eqid 2824 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
6362xkotopon 22211 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6438, 41, 63syl2anc 587 . . 3 (𝜑 → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
65 ovex 7182 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
6665pwex 5268 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
677, 8, 9xkotf 22196 . . . . . 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)
68 frn 6509 . . . . . 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))
6967, 68ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑇)
7066, 69ssexi 5212 . . . 4 ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V
7170a1i 11 . . 3 (𝜑 → ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) ∈ V)
72 cntop2 21852 . . . . 5 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
732, 72syl 17 . . . 4 (𝜑𝑇 ∈ Top)
747, 8, 9xkoval 22198 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
7538, 73, 74syl2anc 587 . . 3 (𝜑 → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
76 eqid 2824 . . . . 5 (𝑇ko 𝑅) = (𝑇ko 𝑅)
7776xkotopon 22211 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
7838, 73, 77syl2anc 587 . . 3 (𝜑 → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
7964, 71, 75, 78subbascn 21865 . 2 (𝜑 → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆ko 𝑅) Cn (𝑇ko 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) ∈ (𝑆ko 𝑅))))
806, 61, 79mpbir2and 712 1 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆ko 𝑅) Cn (𝑇ko 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ∃wrex 3134  {crab 3137  Vcvv 3480   ⊆ wss 3919  𝒫 cpw 4522  ∪ cuni 4824   ↦ cmpt 5132   × cxp 5540  ◡ccnv 5541  dom cdm 5542  ran crn 5543   “ cima 5545   ∘ ccom 5546  Fun wfun 6337  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149   ∈ cmpo 7151  ficfi 8871   ↾t crest 16694  topGenctg 16711  Topctop 21504  TopOnctopon 21521   Cn ccn 21835  Compccmp 21997   ↑ko cxko 22172 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-fin 8509  df-fi 8872  df-rest 16696  df-topgen 16717  df-top 21505  df-topon 21522  df-bases 21557  df-cn 21838  df-cmp 21998  df-xko 22174 This theorem is referenced by:  cnmptk1  22292
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