Proof of Theorem lmbrf
Step | Hyp | Ref
| Expression |
1 | | lmbr.2 |
. . 3
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | lmbr2.4 |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
3 | | lmbr2.5 |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | 1, 2, 3 | lmbr2 22156 |
. 2
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
5 | | 3anass 1097 |
. . 3
⊢ ((𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
6 | 2 | uztrn2 12457 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
7 | | lmbrf.7 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
8 | 7 | eleq1d 2822 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ∈ 𝑢 ↔ 𝐴 ∈ 𝑢)) |
9 | | lmbrf.6 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹:𝑍⟶𝑋) |
10 | 9 | fdmd 6556 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → dom 𝐹 = 𝑍) |
11 | 10 | eleq2d 2823 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ dom 𝐹 ↔ 𝑘 ∈ 𝑍)) |
12 | 11 | biimpar 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ dom 𝐹) |
13 | 12 | biantrurd 536 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
14 | 8, 13 | bitr3d 284 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 ∈ 𝑢 ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
15 | 6, 14 | sylan2 596 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (𝐴 ∈ 𝑢 ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
16 | 15 | anassrs 471 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐴 ∈ 𝑢 ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
17 | 16 | ralbidva 3117 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
18 | 17 | rexbidva 3215 |
. . . . . . 7
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
19 | 18 | imbi2d 344 |
. . . . . 6
⊢ (𝜑 → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢) ↔ (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) |
20 | 19 | ralbidv 3118 |
. . . . 5
⊢ (𝜑 → (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢) ↔ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) |
21 | 20 | anbi2d 632 |
. . . 4
⊢ (𝜑 → ((𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢)) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
22 | | toponmax 21823 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
23 | 1, 22 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝐽) |
24 | | cnex 10810 |
. . . . . . 7
⊢ ℂ
∈ V |
25 | 23, 24 | jctir 524 |
. . . . . 6
⊢ (𝜑 → (𝑋 ∈ 𝐽 ∧ ℂ ∈ V)) |
26 | | uzssz 12459 |
. . . . . . . . 9
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
27 | | zsscn 12184 |
. . . . . . . . 9
⊢ ℤ
⊆ ℂ |
28 | 26, 27 | sstri 3910 |
. . . . . . . 8
⊢
(ℤ≥‘𝑀) ⊆ ℂ |
29 | 2, 28 | eqsstri 3935 |
. . . . . . 7
⊢ 𝑍 ⊆
ℂ |
30 | 9, 29 | jctir 524 |
. . . . . 6
⊢ (𝜑 → (𝐹:𝑍⟶𝑋 ∧ 𝑍 ⊆ ℂ)) |
31 | | elpm2r 8526 |
. . . . . 6
⊢ (((𝑋 ∈ 𝐽 ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶𝑋 ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (𝑋 ↑pm
ℂ)) |
32 | 25, 30, 31 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm
ℂ)) |
33 | 32 | biantrurd 536 |
. . . 4
⊢ (𝜑 → ((𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))) |
34 | 21, 33 | bitr2d 283 |
. . 3
⊢ (𝜑 → ((𝐹 ∈ (𝑋 ↑pm ℂ) ∧ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢)))) |
35 | 5, 34 | syl5bb 286 |
. 2
⊢ (𝜑 → ((𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢)))) |
36 | 4, 35 | bitrd 282 |
1
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢)))) |