| Step | Hyp | Ref
| Expression |
| 1 | | uzf 12881 |
. . . . . . . 8
⊢
ℤ≥:ℤ⟶𝒫 ℤ |
| 2 | | ffn 6736 |
. . . . . . . 8
⊢
(ℤ≥:ℤ⟶𝒫 ℤ →
ℤ≥ Fn ℤ) |
| 3 | 1, 2 | ax-mp 5 |
. . . . . . 7
⊢
ℤ≥ Fn ℤ |
| 4 | | lmflf.1 |
. . . . . . . 8
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 5 | | uzssz 12899 |
. . . . . . . 8
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
| 6 | 4, 5 | eqsstri 4030 |
. . . . . . 7
⊢ 𝑍 ⊆
ℤ |
| 7 | | imaeq2 6074 |
. . . . . . . . 9
⊢ (𝑦 =
(ℤ≥‘𝑗) → (𝐹 “ 𝑦) = (𝐹 “ (ℤ≥‘𝑗))) |
| 8 | 7 | sseq1d 4015 |
. . . . . . . 8
⊢ (𝑦 =
(ℤ≥‘𝑗) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥)) |
| 9 | 8 | rexima 7258 |
. . . . . . 7
⊢
((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ) → (∃𝑦 ∈ (ℤ≥
“ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥)) |
| 10 | 3, 6, 9 | mp2an 692 |
. . . . . 6
⊢
(∃𝑦 ∈
(ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥) |
| 11 | | simpl3 1194 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶𝑋) |
| 12 | 11 | ffund 6740 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → Fun 𝐹) |
| 13 | | uzss 12901 |
. . . . . . . . . . 11
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝑗) ⊆
(ℤ≥‘𝑀)) |
| 14 | 13, 4 | eleq2s 2859 |
. . . . . . . . . 10
⊢ (𝑗 ∈ 𝑍 → (ℤ≥‘𝑗) ⊆
(ℤ≥‘𝑀)) |
| 15 | 14 | adantl 481 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆
(ℤ≥‘𝑀)) |
| 16 | 11 | fdmd 6746 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = 𝑍) |
| 17 | 16, 4 | eqtrdi 2793 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → dom 𝐹 = (ℤ≥‘𝑀)) |
| 18 | 15, 17 | sseqtrrd 4021 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ dom 𝐹) |
| 19 | | funimass4 6973 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧
(ℤ≥‘𝑗) ⊆ dom 𝐹) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥)) |
| 20 | 12, 18, 19 | syl2anc 584 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥)) |
| 21 | 20 | rexbidva 3177 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∃𝑗 ∈ 𝑍 (𝐹 “ (ℤ≥‘𝑗)) ⊆ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥)) |
| 22 | 10, 21 | bitr2id 284 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑥 ↔ ∃𝑦 ∈ (ℤ≥ “ 𝑍)(𝐹 “ 𝑦) ⊆ 𝑥)) |
| 23 | 22 | imbi2d 340 |
. . . 4
⊢ ((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))) |
| 24 | 23 | ralbidv 3178 |
. . 3
⊢ ((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))) |
| 25 | 24 | anbi2d 630 |
. 2
⊢ ((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)))) |
| 26 | | simp1 1137 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
| 27 | | simp2 1138 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝑀 ∈ ℤ) |
| 28 | | simp3 1139 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → 𝐹:𝑍⟶𝑋) |
| 29 | | eqidd 2738 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘)) |
| 30 | 26, 4, 27, 28, 29 | lmbrf 23268 |
. 2
⊢ ((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)))) |
| 31 | 4 | uzfbas 23906 |
. . 3
⊢ (𝑀 ∈ ℤ →
(ℤ≥ “ 𝑍) ∈ (fBas‘𝑍)) |
| 32 | | lmflf.2 |
. . . 4
⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍)) |
| 33 | 32 | flffbas 24003 |
. . 3
⊢ ((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 | 31, 33 | syl3an2 1165 |
. 2
⊢ ((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 | 25, 30, 34 | 3bitr4d 311 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹))) |