Step |
Hyp |
Ref |
Expression |
1 |
|
lmflf.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
lmflf.2 |
⊢ 𝐿 = ( 𝑍 filGen ( ℤ≥ “ 𝑍 ) ) |
3 |
|
uzf |
⊢ ℤ≥ : ℤ ⟶ 𝒫 ℤ |
4 |
|
ffn |
⊢ ( ℤ≥ : ℤ ⟶ 𝒫 ℤ → ℤ≥ Fn ℤ ) |
5 |
3 4
|
ax-mp |
⊢ ℤ≥ Fn ℤ |
6 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
7 |
1 6
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
8 |
|
imaeq2 |
⊢ ( 𝑦 = ( ℤ≥ ‘ 𝑗 ) → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ) |
9 |
8
|
sseq1d |
⊢ ( 𝑦 = ( ℤ≥ ‘ 𝑗 ) → ( ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ↔ ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ⊆ 𝑥 ) ) |
10 |
9
|
rexima |
⊢ ( ( ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ ) → ( ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ⊆ 𝑥 ) ) |
11 |
5 7 10
|
mp2an |
⊢ ( ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ⊆ 𝑥 ) |
12 |
|
simpl3 |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → 𝐹 : 𝑍 ⟶ 𝑋 ) |
13 |
12
|
ffund |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → Fun 𝐹 ) |
14 |
|
uzss |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ℤ≥ ‘ 𝑗 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
15 |
14 1
|
eleq2s |
⊢ ( 𝑗 ∈ 𝑍 → ( ℤ≥ ‘ 𝑗 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
16 |
15
|
adantl |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑗 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
17 |
12
|
fdmd |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → dom 𝐹 = 𝑍 ) |
18 |
17 1
|
eqtrdi |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → dom 𝐹 = ( ℤ≥ ‘ 𝑀 ) ) |
19 |
16 18
|
sseqtrrd |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑗 ) ⊆ dom 𝐹 ) |
20 |
|
funimass4 |
⊢ ( ( Fun 𝐹 ∧ ( ℤ≥ ‘ 𝑗 ) ⊆ dom 𝐹 ) → ( ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ⊆ 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ) |
21 |
13 19 20
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ⊆ 𝑥 ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ) |
22 |
21
|
rexbidva |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( ∃ 𝑗 ∈ 𝑍 ( 𝐹 “ ( ℤ≥ ‘ 𝑗 ) ) ⊆ 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ) |
23 |
11 22
|
bitr2id |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ↔ ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ) ) |
24 |
23
|
imbi2d |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( ( 𝑃 ∈ 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ↔ ( 𝑃 ∈ 𝑥 → ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ) ) ) |
25 |
24
|
ralbidv |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ) ) ) |
26 |
25
|
anbi2d |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ) ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ) ) ) ) |
27 |
|
simp1 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
28 |
|
simp2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → 𝑀 ∈ ℤ ) |
29 |
|
simp3 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → 𝐹 : 𝑍 ⟶ 𝑋 ) |
30 |
|
eqidd |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
31 |
27 1 28 29 30
|
lmbrf |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ 𝑥 ) ) ) ) |
32 |
1
|
uzfbas |
⊢ ( 𝑀 ∈ ℤ → ( ℤ≥ “ 𝑍 ) ∈ ( fBas ‘ 𝑍 ) ) |
33 |
2
|
flffbas |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ ( ℤ≥ “ 𝑍 ) ∈ ( fBas ‘ 𝑍 ) ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( 𝑃 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ) ) ) ) |
34 |
32 33
|
syl3an2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( 𝑃 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝑃 ∈ 𝑋 ∧ ∀ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 → ∃ 𝑦 ∈ ( ℤ≥ “ 𝑍 ) ( 𝐹 “ 𝑦 ) ⊆ 𝑥 ) ) ) ) |
35 |
26 31 34
|
3bitr4d |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑀 ∈ ℤ ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ 𝑃 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ) ) |