Step |
Hyp |
Ref |
Expression |
1 |
|
xkoco2cn.r |
⊢ ( 𝜑 → 𝑅 ∈ Top ) |
2 |
|
xkoco2cn.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑆 Cn 𝑇 ) ) |
3 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) |
4 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → 𝐹 ∈ ( 𝑆 Cn 𝑇 ) ) |
5 |
|
cnco |
⊢ ( ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∧ 𝐹 ∈ ( 𝑆 Cn 𝑇 ) ) → ( 𝐹 ∘ 𝑔 ) ∈ ( 𝑅 Cn 𝑇 ) ) |
6 |
3 4 5
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( 𝐹 ∘ 𝑔 ) ∈ ( 𝑅 Cn 𝑇 ) ) |
7 |
6
|
fmpttd |
⊢ ( 𝜑 → ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) : ( 𝑅 Cn 𝑆 ) ⟶ ( 𝑅 Cn 𝑇 ) ) |
8 |
|
eqid |
⊢ ∪ 𝑅 = ∪ 𝑅 |
9 |
|
eqid |
⊢ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } = { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } |
10 |
|
eqid |
⊢ ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) = ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) |
11 |
8 9 10
|
xkobval |
⊢ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) = { 𝑥 ∣ ∃ 𝑘 ∈ 𝒫 ∪ 𝑅 ∃ 𝑣 ∈ 𝑇 ( ( 𝑅 ↾t 𝑘 ) ∈ Comp ∧ 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) } |
12 |
11
|
abeq2i |
⊢ ( 𝑥 ∈ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ↔ ∃ 𝑘 ∈ 𝒫 ∪ 𝑅 ∃ 𝑣 ∈ 𝑇 ( ( 𝑅 ↾t 𝑘 ) ∈ Comp ∧ 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ) |
13 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) |
14 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → 𝐹 ∈ ( 𝑆 Cn 𝑇 ) ) |
15 |
13 14 5
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( 𝐹 ∘ 𝑔 ) ∈ ( 𝑅 Cn 𝑇 ) ) |
16 |
|
imaeq1 |
⊢ ( ℎ = ( 𝐹 ∘ 𝑔 ) → ( ℎ “ 𝑘 ) = ( ( 𝐹 ∘ 𝑔 ) “ 𝑘 ) ) |
17 |
|
imaco |
⊢ ( ( 𝐹 ∘ 𝑔 ) “ 𝑘 ) = ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) |
18 |
16 17
|
eqtrdi |
⊢ ( ℎ = ( 𝐹 ∘ 𝑔 ) → ( ℎ “ 𝑘 ) = ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) ) |
19 |
18
|
sseq1d |
⊢ ( ℎ = ( 𝐹 ∘ 𝑔 ) → ( ( ℎ “ 𝑘 ) ⊆ 𝑣 ↔ ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) ⊆ 𝑣 ) ) |
20 |
19
|
elrab3 |
⊢ ( ( 𝐹 ∘ 𝑔 ) ∈ ( 𝑅 Cn 𝑇 ) → ( ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ↔ ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) ⊆ 𝑣 ) ) |
21 |
15 20
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ↔ ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) ⊆ 𝑣 ) ) |
22 |
|
eqid |
⊢ ∪ 𝑆 = ∪ 𝑆 |
23 |
|
eqid |
⊢ ∪ 𝑇 = ∪ 𝑇 |
24 |
22 23
|
cnf |
⊢ ( 𝐹 ∈ ( 𝑆 Cn 𝑇 ) → 𝐹 : ∪ 𝑆 ⟶ ∪ 𝑇 ) |
25 |
2 24
|
syl |
⊢ ( 𝜑 → 𝐹 : ∪ 𝑆 ⟶ ∪ 𝑇 ) |
26 |
25
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → 𝐹 : ∪ 𝑆 ⟶ ∪ 𝑇 ) |
27 |
26
|
ffund |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → Fun 𝐹 ) |
28 |
|
imassrn |
⊢ ( 𝑔 “ 𝑘 ) ⊆ ran 𝑔 |
29 |
8 22
|
cnf |
⊢ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) → 𝑔 : ∪ 𝑅 ⟶ ∪ 𝑆 ) |
30 |
13 29
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → 𝑔 : ∪ 𝑅 ⟶ ∪ 𝑆 ) |
31 |
30
|
frnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ran 𝑔 ⊆ ∪ 𝑆 ) |
32 |
28 31
|
sstrid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( 𝑔 “ 𝑘 ) ⊆ ∪ 𝑆 ) |
33 |
26
|
fdmd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → dom 𝐹 = ∪ 𝑆 ) |
34 |
32 33
|
sseqtrrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( 𝑔 “ 𝑘 ) ⊆ dom 𝐹 ) |
35 |
|
funimass3 |
⊢ ( ( Fun 𝐹 ∧ ( 𝑔 “ 𝑘 ) ⊆ dom 𝐹 ) → ( ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) ⊆ 𝑣 ↔ ( 𝑔 “ 𝑘 ) ⊆ ( ◡ 𝐹 “ 𝑣 ) ) ) |
36 |
27 34 35
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( ( 𝐹 “ ( 𝑔 “ 𝑘 ) ) ⊆ 𝑣 ↔ ( 𝑔 “ 𝑘 ) ⊆ ( ◡ 𝐹 “ 𝑣 ) ) ) |
37 |
21 36
|
bitrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) ∧ 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ) → ( ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ↔ ( 𝑔 “ 𝑘 ) ⊆ ( ◡ 𝐹 “ 𝑣 ) ) ) |
38 |
37
|
rabbidva |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } } = { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑔 “ 𝑘 ) ⊆ ( ◡ 𝐹 “ 𝑣 ) } ) |
39 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → 𝑅 ∈ Top ) |
40 |
|
cntop1 |
⊢ ( 𝐹 ∈ ( 𝑆 Cn 𝑇 ) → 𝑆 ∈ Top ) |
41 |
2 40
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Top ) |
42 |
41
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → 𝑆 ∈ Top ) |
43 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → 𝑘 ∈ 𝒫 ∪ 𝑅 ) |
44 |
43
|
elpwid |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → 𝑘 ⊆ ∪ 𝑅 ) |
45 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → ( 𝑅 ↾t 𝑘 ) ∈ Comp ) |
46 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → 𝐹 ∈ ( 𝑆 Cn 𝑇 ) ) |
47 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → 𝑣 ∈ 𝑇 ) |
48 |
|
cnima |
⊢ ( ( 𝐹 ∈ ( 𝑆 Cn 𝑇 ) ∧ 𝑣 ∈ 𝑇 ) → ( ◡ 𝐹 “ 𝑣 ) ∈ 𝑆 ) |
49 |
46 47 48
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → ( ◡ 𝐹 “ 𝑣 ) ∈ 𝑆 ) |
50 |
8 39 42 44 45 49
|
xkoopn |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑔 “ 𝑘 ) ⊆ ( ◡ 𝐹 “ 𝑣 ) } ∈ ( 𝑆 ↑ko 𝑅 ) ) |
51 |
38 50
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } } ∈ ( 𝑆 ↑ko 𝑅 ) ) |
52 |
|
imaeq2 |
⊢ ( 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } → ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) = ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ) |
53 |
|
eqid |
⊢ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) = ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) |
54 |
53
|
mptpreima |
⊢ ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) = { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } } |
55 |
52 54
|
eqtrdi |
⊢ ( 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } → ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) = { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } } ) |
56 |
55
|
eleq1d |
⊢ ( 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } → ( ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ↔ { 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝐹 ∘ 𝑔 ) ∈ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } } ∈ ( 𝑆 ↑ko 𝑅 ) ) ) |
57 |
51 56
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) ∧ ( 𝑅 ↾t 𝑘 ) ∈ Comp ) → ( 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } → ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ) ) |
58 |
57
|
expimpd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑇 ) ) → ( ( ( 𝑅 ↾t 𝑘 ) ∈ Comp ∧ 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) → ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ) ) |
59 |
58
|
rexlimdvva |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝒫 ∪ 𝑅 ∃ 𝑣 ∈ 𝑇 ( ( 𝑅 ↾t 𝑘 ) ∈ Comp ∧ 𝑥 = { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) → ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ) ) |
60 |
12 59
|
syl5bi |
⊢ ( 𝜑 → ( 𝑥 ∈ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) → ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ) ) |
61 |
60
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ) |
62 |
|
eqid |
⊢ ( 𝑆 ↑ko 𝑅 ) = ( 𝑆 ↑ko 𝑅 ) |
63 |
62
|
xkotopon |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) ∈ ( TopOn ‘ ( 𝑅 Cn 𝑆 ) ) ) |
64 |
1 41 63
|
syl2anc |
⊢ ( 𝜑 → ( 𝑆 ↑ko 𝑅 ) ∈ ( TopOn ‘ ( 𝑅 Cn 𝑆 ) ) ) |
65 |
|
ovex |
⊢ ( 𝑅 Cn 𝑇 ) ∈ V |
66 |
65
|
pwex |
⊢ 𝒫 ( 𝑅 Cn 𝑇 ) ∈ V |
67 |
8 9 10
|
xkotf |
⊢ ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) : ( { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } × 𝑇 ) ⟶ 𝒫 ( 𝑅 Cn 𝑇 ) |
68 |
|
frn |
⊢ ( ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) : ( { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } × 𝑇 ) ⟶ 𝒫 ( 𝑅 Cn 𝑇 ) → ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ⊆ 𝒫 ( 𝑅 Cn 𝑇 ) ) |
69 |
67 68
|
ax-mp |
⊢ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ⊆ 𝒫 ( 𝑅 Cn 𝑇 ) |
70 |
66 69
|
ssexi |
⊢ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ∈ V |
71 |
70
|
a1i |
⊢ ( 𝜑 → ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ∈ V ) |
72 |
|
cntop2 |
⊢ ( 𝐹 ∈ ( 𝑆 Cn 𝑇 ) → 𝑇 ∈ Top ) |
73 |
2 72
|
syl |
⊢ ( 𝜑 → 𝑇 ∈ Top ) |
74 |
8 9 10
|
xkoval |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑇 ∈ Top ) → ( 𝑇 ↑ko 𝑅 ) = ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ) ) ) |
75 |
1 73 74
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 ↑ko 𝑅 ) = ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ) ) ) |
76 |
|
eqid |
⊢ ( 𝑇 ↑ko 𝑅 ) = ( 𝑇 ↑ko 𝑅 ) |
77 |
76
|
xkotopon |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑇 ∈ Top ) → ( 𝑇 ↑ko 𝑅 ) ∈ ( TopOn ‘ ( 𝑅 Cn 𝑇 ) ) ) |
78 |
1 73 77
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 ↑ko 𝑅 ) ∈ ( TopOn ‘ ( 𝑅 Cn 𝑇 ) ) ) |
79 |
64 71 75 78
|
subbascn |
⊢ ( 𝜑 → ( ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) ∈ ( ( 𝑆 ↑ko 𝑅 ) Cn ( 𝑇 ↑ko 𝑅 ) ) ↔ ( ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) : ( 𝑅 Cn 𝑆 ) ⟶ ( 𝑅 Cn 𝑇 ) ∧ ∀ 𝑥 ∈ ran ( 𝑘 ∈ { 𝑦 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑦 ) ∈ Comp } , 𝑣 ∈ 𝑇 ↦ { ℎ ∈ ( 𝑅 Cn 𝑇 ) ∣ ( ℎ “ 𝑘 ) ⊆ 𝑣 } ) ( ◡ ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) “ 𝑥 ) ∈ ( 𝑆 ↑ko 𝑅 ) ) ) ) |
80 |
7 61 79
|
mpbir2and |
⊢ ( 𝜑 → ( 𝑔 ∈ ( 𝑅 Cn 𝑆 ) ↦ ( 𝐹 ∘ 𝑔 ) ) ∈ ( ( 𝑆 ↑ko 𝑅 ) Cn ( 𝑇 ↑ko 𝑅 ) ) ) |