Step |
Hyp |
Ref |
Expression |
1 |
|
cnneiima.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
2 |
|
cnneiima.2 |
⊢ ( 𝜑 → 𝑁 ∈ ( ( nei ‘ 𝐾 ) ‘ 𝑇 ) ) |
3 |
|
cnneiima.3 |
⊢ ( 𝜑 → 𝑆 ⊆ ( ◡ 𝐹 “ 𝑇 ) ) |
4 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
5 |
|
eqid |
⊢ ∪ 𝐾 = ∪ 𝐾 |
6 |
4 5
|
cnf |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) |
8 |
7
|
ffund |
⊢ ( 𝜑 → Fun 𝐹 ) |
9 |
|
cntop2 |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐾 ∈ Top ) |
10 |
1 9
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ Top ) |
11 |
5
|
neiss2 |
⊢ ( ( 𝐾 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐾 ) ‘ 𝑇 ) ) → 𝑇 ⊆ ∪ 𝐾 ) |
12 |
10 2 11
|
syl2anc |
⊢ ( 𝜑 → 𝑇 ⊆ ∪ 𝐾 ) |
13 |
5
|
neii1 |
⊢ ( ( 𝐾 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐾 ) ‘ 𝑇 ) ) → 𝑁 ⊆ ∪ 𝐾 ) |
14 |
10 2 13
|
syl2anc |
⊢ ( 𝜑 → 𝑁 ⊆ ∪ 𝐾 ) |
15 |
5
|
neiint |
⊢ ( ( 𝐾 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐾 ∧ 𝑁 ⊆ ∪ 𝐾 ) → ( 𝑁 ∈ ( ( nei ‘ 𝐾 ) ‘ 𝑇 ) ↔ 𝑇 ⊆ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ) |
16 |
10 12 14 15
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ∈ ( ( nei ‘ 𝐾 ) ‘ 𝑇 ) ↔ 𝑇 ⊆ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ) |
17 |
2 16
|
mpbid |
⊢ ( 𝜑 → 𝑇 ⊆ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) |
18 |
|
sspreima |
⊢ ( ( Fun 𝐹 ∧ 𝑇 ⊆ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) → ( ◡ 𝐹 “ 𝑇 ) ⊆ ( ◡ 𝐹 “ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ) |
19 |
8 17 18
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑇 ) ⊆ ( ◡ 𝐹 “ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ) |
20 |
3 19
|
sstrd |
⊢ ( 𝜑 → 𝑆 ⊆ ( ◡ 𝐹 “ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ) |
21 |
5
|
cnntri |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝑁 ⊆ ∪ 𝐾 ) → ( ◡ 𝐹 “ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ⊆ ( ( int ‘ 𝐽 ) ‘ ( ◡ 𝐹 “ 𝑁 ) ) ) |
22 |
1 14 21
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( ( int ‘ 𝐾 ) ‘ 𝑁 ) ) ⊆ ( ( int ‘ 𝐽 ) ‘ ( ◡ 𝐹 “ 𝑁 ) ) ) |
23 |
20 22
|
sstrd |
⊢ ( 𝜑 → 𝑆 ⊆ ( ( int ‘ 𝐽 ) ‘ ( ◡ 𝐹 “ 𝑁 ) ) ) |
24 |
|
cntop1 |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐽 ∈ Top ) |
25 |
1 24
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
26 |
|
sspreima |
⊢ ( ( Fun 𝐹 ∧ 𝑇 ⊆ ∪ 𝐾 ) → ( ◡ 𝐹 “ 𝑇 ) ⊆ ( ◡ 𝐹 “ ∪ 𝐾 ) ) |
27 |
8 12 26
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑇 ) ⊆ ( ◡ 𝐹 “ ∪ 𝐾 ) ) |
28 |
|
fimacnv |
⊢ ( 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 → ( ◡ 𝐹 “ ∪ 𝐾 ) = ∪ 𝐽 ) |
29 |
7 28
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ∪ 𝐾 ) = ∪ 𝐽 ) |
30 |
27 29
|
sseqtrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑇 ) ⊆ ∪ 𝐽 ) |
31 |
3 30
|
sstrd |
⊢ ( 𝜑 → 𝑆 ⊆ ∪ 𝐽 ) |
32 |
|
sspreima |
⊢ ( ( Fun 𝐹 ∧ 𝑁 ⊆ ∪ 𝐾 ) → ( ◡ 𝐹 “ 𝑁 ) ⊆ ( ◡ 𝐹 “ ∪ 𝐾 ) ) |
33 |
8 14 32
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑁 ) ⊆ ( ◡ 𝐹 “ ∪ 𝐾 ) ) |
34 |
33 29
|
sseqtrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑁 ) ⊆ ∪ 𝐽 ) |
35 |
4
|
neiint |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ ( ◡ 𝐹 “ 𝑁 ) ⊆ ∪ 𝐽 ) → ( ( ◡ 𝐹 “ 𝑁 ) ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑆 ⊆ ( ( int ‘ 𝐽 ) ‘ ( ◡ 𝐹 “ 𝑁 ) ) ) ) |
36 |
25 31 34 35
|
syl3anc |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ 𝑁 ) ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑆 ⊆ ( ( int ‘ 𝐽 ) ‘ ( ◡ 𝐹 “ 𝑁 ) ) ) ) |
37 |
23 36
|
mpbird |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝑁 ) ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) |