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Theorem xkoco2cn 22194
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)) ∈ ((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 485 . . . 4 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
2 xkoco2cn.f . . . . 5 (𝜑𝐹 ∈ (𝑆 Cn 𝑇))
32adantr 481 . . . 4 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
4 cnco 21802 . . . 4 ((𝑔 ∈ (𝑅 Cn 𝑆) ∧ 𝐹 ∈ (𝑆 Cn 𝑇)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
51, 3, 4syl2anc 584 . . 3 ((𝜑𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
65fmpttd 6871 . 2 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)):(𝑅 Cn 𝑆)⟶(𝑅 Cn 𝑇))
7 eqid 2818 . . . . . 6 𝑅 = 𝑅
8 eqid 2818 . . . . . 6 {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp} = {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}
9 eqid 2818 . . . . . 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 22122 . . . . 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 2945 . . . 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 485 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝑔 ∈ (𝑅 Cn 𝑆))
132ad3antrrr 726 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹 ∈ (𝑆 Cn 𝑇))
1412, 13, 4syl2anc 584 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝐹𝑔) ∈ (𝑅 Cn 𝑇))
15 imaeq1 5917 . . . . . . . . . . . . . 14 ( = (𝐹𝑔) → (𝑘) = ((𝐹𝑔) “ 𝑘))
16 imaco 6097 . . . . . . . . . . . . . 14 ((𝐹𝑔) “ 𝑘) = (𝐹 “ (𝑔𝑘))
1715, 16syl6eq 2869 . . . . . . . . . . . . 13 ( = (𝐹𝑔) → (𝑘) = (𝐹 “ (𝑔𝑘)))
1817sseq1d 3995 . . . . . . . . . . . 12 ( = (𝐹𝑔) → ((𝑘) ⊆ 𝑣 ↔ (𝐹 “ (𝑔𝑘)) ⊆ 𝑣))
1918elrab3 3678 . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . 15 𝑆 = 𝑆
22 eqid 2818 . . . . . . . . . . . . . . 15 𝑇 = 𝑇
2321, 22cnf 21782 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝐹: 𝑆 𝑇)
242, 23syl 17 . . . . . . . . . . . . 13 (𝜑𝐹: 𝑆 𝑇)
2524ad3antrrr 726 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → 𝐹: 𝑆 𝑇)
2625ffund 6511 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → Fun 𝐹)
27 imassrn 5933 . . . . . . . . . . . . 13 (𝑔𝑘) ⊆ ran 𝑔
287, 21cnf 21782 . . . . . . . . . . . . . . 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 6514 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ran 𝑔 𝑆)
3127, 30sstrid 3975 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ 𝑆)
3225fdmd 6516 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → dom 𝐹 = 𝑆)
3331, 32sseqtrrd 4005 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑔𝑘) ⊆ dom 𝐹)
34 funimass3 6816 . . . . . . . . . . 11 ((Fun 𝐹 ∧ (𝑔𝑘) ⊆ dom 𝐹) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3526, 33, 34syl2anc 584 . . . . . . . . . 10 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹 “ (𝑔𝑘)) ⊆ 𝑣 ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3620, 35bitrd 280 . . . . . . . . 9 ((((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → ((𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} ↔ (𝑔𝑘) ⊆ (𝐹𝑣)))
3736rabbidva 3476 . . . . . . . 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 722 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑅 ∈ Top)
40 cntop1 21776 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑆 ∈ Top)
412, 40syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Top)
4241ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ Top)
43 simplrl 773 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
4443elpwid 4549 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 𝑅)
45 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑅t 𝑘) ∈ Comp)
462ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝐹 ∈ (𝑆 Cn 𝑇))
47 simplrr 774 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑇)
48 cnima 21801 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 Cn 𝑇) ∧ 𝑣𝑇) → (𝐹𝑣) ∈ 𝑆)
4946, 47, 48syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝐹𝑣) ∈ 𝑆)
507, 39, 42, 44, 45, 49xkoopn 22125 . . . . . . . 8 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝑔𝑘) ⊆ (𝐹𝑣)} ∈ (𝑆ko 𝑅))
5137, 50eqeltrd 2910 . . . . . . 7 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}} ∈ (𝑆ko 𝑅))
52 imaeq2 5918 . . . . . . . . 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 2818 . . . . . . . . . 10 (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔))
5453mptpreima 6085 . . . . . . . . 9 ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}}
5552, 54syl6eq 2869 . . . . . . . 8 (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) = {𝑔 ∈ (𝑅 Cn 𝑆) ∣ (𝐹𝑔) ∈ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}})
5655eleq1d 2894 . . . . . . 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 248 . . . . . 6 (((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣} → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
5857expimpd 454 . . . . 5 ((𝜑 ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑇)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
5958rexlimdvva 3291 . . . 4 (𝜑 → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑇 ((𝑅t 𝑘) ∈ Comp ∧ 𝑥 = { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}) → ((𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) “ 𝑥) ∈ (𝑆ko 𝑅)))
6011, 59syl5bi 243 . . 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 3178 . 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 2818 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
6362xkotopon 22136 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
6438, 41, 63syl2anc 584 . . 3 (𝜑 → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
65 ovex 7178 . . . . . 6 (𝑅 Cn 𝑇) ∈ V
6665pwex 5272 . . . . 5 𝒫 (𝑅 Cn 𝑇) ∈ V
677, 8, 9xkotf 22121 . . . . . 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 6513 . . . . . 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 5217 . . . 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 21777 . . . . 5 (𝐹 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top)
732, 72syl 17 . . . 4 (𝜑𝑇 ∈ Top)
747, 8, 9xkoval 22123 . . . 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 584 . . 3 (𝜑 → (𝑇ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑦 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑦) ∈ Comp}, 𝑣𝑇 ↦ { ∈ (𝑅 Cn 𝑇) ∣ (𝑘) ⊆ 𝑣}))))
76 eqid 2818 . . . . 5 (𝑇ko 𝑅) = (𝑇ko 𝑅)
7776xkotopon 22136 . . . 4 ((𝑅 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
7838, 73, 77syl2anc 584 . . 3 (𝜑 → (𝑇ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑇)))
7964, 71, 75, 78subbascn 21790 . 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 709 1 (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆ko 𝑅) Cn (𝑇ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  ficfi 8862  t crest 16682  topGenctg 16699  Topctop 21429  TopOnctopon 21446   Cn ccn 21760  Compccmp 21922  ko cxko 22097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-cmp 21923  df-xko 22099
This theorem is referenced by:  cnmptk1  22217
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