Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐽 ∈ Comp ) |
2 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
3 |
|
eqid |
⊢ ∪ 𝐾 = ∪ 𝐾 |
4 |
2 3
|
cnf |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) |
5 |
4
|
adantl |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) |
6 |
5
|
ffnd |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 Fn ∪ 𝐽 ) |
7 |
|
dffn4 |
⊢ ( 𝐹 Fn ∪ 𝐽 ↔ 𝐹 : ∪ 𝐽 –onto→ ran 𝐹 ) |
8 |
6 7
|
sylib |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : ∪ 𝐽 –onto→ ran 𝐹 ) |
9 |
|
cntop2 |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐾 ∈ Top ) |
10 |
9
|
adantl |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐾 ∈ Top ) |
11 |
5
|
frnd |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ran 𝐹 ⊆ ∪ 𝐾 ) |
12 |
3
|
restuni |
⊢ ( ( 𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾 ) → ran 𝐹 = ∪ ( 𝐾 ↾t ran 𝐹 ) ) |
13 |
10 11 12
|
syl2anc |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ran 𝐹 = ∪ ( 𝐾 ↾t ran 𝐹 ) ) |
14 |
|
foeq3 |
⊢ ( ran 𝐹 = ∪ ( 𝐾 ↾t ran 𝐹 ) → ( 𝐹 : ∪ 𝐽 –onto→ ran 𝐹 ↔ 𝐹 : ∪ 𝐽 –onto→ ∪ ( 𝐾 ↾t ran 𝐹 ) ) ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝐹 : ∪ 𝐽 –onto→ ran 𝐹 ↔ 𝐹 : ∪ 𝐽 –onto→ ∪ ( 𝐾 ↾t ran 𝐹 ) ) ) |
16 |
8 15
|
mpbid |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : ∪ 𝐽 –onto→ ∪ ( 𝐾 ↾t ran 𝐹 ) ) |
17 |
|
simpr |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
18 |
|
toptopon2 |
⊢ ( 𝐾 ∈ Top ↔ 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ) |
19 |
10 18
|
sylib |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ) |
20 |
|
ssidd |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ran 𝐹 ⊆ ran 𝐹 ) |
21 |
|
cnrest2 |
⊢ ( ( 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ ∪ 𝐾 ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ 𝐹 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ) ) |
22 |
19 20 11 21
|
syl3anc |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ 𝐹 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ) ) |
23 |
17 22
|
mpbid |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ) |
24 |
|
eqid |
⊢ ∪ ( 𝐾 ↾t ran 𝐹 ) = ∪ ( 𝐾 ↾t ran 𝐹 ) |
25 |
24
|
cncmp |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 : ∪ 𝐽 –onto→ ∪ ( 𝐾 ↾t ran 𝐹 ) ∧ 𝐹 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ) → ( 𝐾 ↾t ran 𝐹 ) ∈ Comp ) |
26 |
1 16 23 25
|
syl3anc |
⊢ ( ( 𝐽 ∈ Comp ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝐾 ↾t ran 𝐹 ) ∈ Comp ) |