Step | Hyp | Ref
| Expression |
1 | | lmcnp.4 |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃)) |
2 | | eqid 2610 |
. . . . . . 7
⊢ ∪ 𝐽 =
∪ 𝐽 |
3 | | eqid 2610 |
. . . . . . 7
⊢ ∪ 𝐾 =
∪ 𝐾 |
4 | 2, 3 | cnpf 20861 |
. . . . . 6
⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐺:∪ 𝐽⟶∪ 𝐾) |
5 | 1, 4 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾) |
6 | | lmcnp.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) |
7 | | cnptop1 20856 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top) |
8 | 1, 7 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ Top) |
9 | 2 | toptopon 20548 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
10 | 8, 9 | sylib 207 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
11 | | nnuz 11599 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
12 | | 1zzd 11285 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℤ) |
13 | 10, 11, 12 | lmbr2 20873 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (∪ 𝐽 ↑pm
ℂ) ∧ 𝑃 ∈
∪ 𝐽 ∧ ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣))))) |
14 | 6, 13 | mpbid 221 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 ∈ (∪ 𝐽 ↑pm
ℂ) ∧ 𝑃 ∈
∪ 𝐽 ∧ ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)))) |
15 | 14 | simp1d 1066 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ (∪ 𝐽 ↑pm
ℂ)) |
16 | | uniexg 6853 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → ∪ 𝐽
∈ V) |
17 | 8, 16 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝐽
∈ V) |
18 | | cnex 9896 |
. . . . . . . 8
⊢ ℂ
∈ V |
19 | | elpm2g 7760 |
. . . . . . . 8
⊢ ((∪ 𝐽
∈ V ∧ ℂ ∈ V) → (𝐹 ∈ (∪ 𝐽 ↑pm
ℂ) ↔ (𝐹:dom
𝐹⟶∪ 𝐽
∧ dom 𝐹 ⊆
ℂ))) |
20 | 17, 18, 19 | sylancl 693 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ∈ (∪ 𝐽 ↑pm
ℂ) ↔ (𝐹:dom
𝐹⟶∪ 𝐽
∧ dom 𝐹 ⊆
ℂ))) |
21 | 15, 20 | mpbid 221 |
. . . . . 6
⊢ (𝜑 → (𝐹:dom 𝐹⟶∪ 𝐽 ∧ dom 𝐹 ⊆ ℂ)) |
22 | 21 | simpld 474 |
. . . . 5
⊢ (𝜑 → 𝐹:dom 𝐹⟶∪ 𝐽) |
23 | | fco 5971 |
. . . . 5
⊢ ((𝐺:∪
𝐽⟶∪ 𝐾
∧ 𝐹:dom 𝐹⟶∪ 𝐽)
→ (𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾) |
24 | 5, 22, 23 | syl2anc 691 |
. . . 4
⊢ (𝜑 → (𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾) |
25 | | fdm 5964 |
. . . . . 6
⊢ ((𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾 → dom (𝐺 ∘ 𝐹) = dom 𝐹) |
26 | 24, 25 | syl 17 |
. . . . 5
⊢ (𝜑 → dom (𝐺 ∘ 𝐹) = dom 𝐹) |
27 | 26 | feq2d 5944 |
. . . 4
⊢ (𝜑 → ((𝐺 ∘ 𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾 ↔ (𝐺 ∘ 𝐹):dom 𝐹⟶∪ 𝐾)) |
28 | 24, 27 | mpbird 246 |
. . 3
⊢ (𝜑 → (𝐺 ∘ 𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾) |
29 | 21 | simprd 478 |
. . . 4
⊢ (𝜑 → dom 𝐹 ⊆ ℂ) |
30 | 26, 29 | eqsstrd 3602 |
. . 3
⊢ (𝜑 → dom (𝐺 ∘ 𝐹) ⊆ ℂ) |
31 | | cnptop2 20857 |
. . . . . 6
⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top) |
32 | 1, 31 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ Top) |
33 | | uniexg 6853 |
. . . . 5
⊢ (𝐾 ∈ Top → ∪ 𝐾
∈ V) |
34 | 32, 33 | syl 17 |
. . . 4
⊢ (𝜑 → ∪ 𝐾
∈ V) |
35 | | elpm2g 7760 |
. . . 4
⊢ ((∪ 𝐾
∈ V ∧ ℂ ∈ V) → ((𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm
ℂ) ↔ ((𝐺 ∘
𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾 ∧ dom (𝐺 ∘ 𝐹) ⊆ ℂ))) |
36 | 34, 18, 35 | sylancl 693 |
. . 3
⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm
ℂ) ↔ ((𝐺 ∘
𝐹):dom (𝐺 ∘ 𝐹)⟶∪ 𝐾 ∧ dom (𝐺 ∘ 𝐹) ⊆ ℂ))) |
37 | 28, 30, 36 | mpbir2and 959 |
. 2
⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm
ℂ)) |
38 | 14 | simp2d 1067 |
. . 3
⊢ (𝜑 → 𝑃 ∈ ∪ 𝐽) |
39 | 5, 38 | ffvelrnd 6268 |
. 2
⊢ (𝜑 → (𝐺‘𝑃) ∈ ∪ 𝐾) |
40 | 14 | simp3d 1068 |
. . . . . 6
⊢ (𝜑 → ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣))) |
41 | 40 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣))) |
42 | | cnpimaex 20870 |
. . . . . . 7
⊢ ((𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) |
43 | 42 | 3expb 1258 |
. . . . . 6
⊢ ((𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) |
44 | 1, 43 | sylan 487 |
. . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) |
45 | | r19.29 3054 |
. . . . . . 7
⊢
((∀𝑣 ∈
𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢))) |
46 | | pm3.45 875 |
. . . . . . . . 9
⊢ ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) → ((𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢))) |
47 | 46 | imp 444 |
. . . . . . . 8
⊢ (((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) |
48 | 47 | reximi 2994 |
. . . . . . 7
⊢
(∃𝑣 ∈
𝐽 ((𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) |
49 | 45, 48 | syl 17 |
. . . . . 6
⊢
((∀𝑣 ∈
𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) |
50 | 5 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝐺:∪ 𝐽⟶∪ 𝐾) |
51 | | ffn 5958 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺:∪
𝐽⟶∪ 𝐾
→ 𝐺 Fn ∪ 𝐽) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝐺 Fn ∪ 𝐽) |
53 | | simplrl 796 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑣 ∈ 𝐽) |
54 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ 𝐽 → 𝑣 ⊆ ∪ 𝐽) |
55 | 53, 54 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑣 ⊆ ∪ 𝐽) |
56 | | fnfvima 6400 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐺 Fn ∪
𝐽 ∧ 𝑣 ⊆ ∪ 𝐽 ∧ (𝐹‘𝑘) ∈ 𝑣) → (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣)) |
57 | 56 | 3expia 1259 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺 Fn ∪
𝐽 ∧ 𝑣 ⊆ ∪ 𝐽) → ((𝐹‘𝑘) ∈ 𝑣 → (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣))) |
58 | 52, 55, 57 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣))) |
59 | 22 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → 𝐹:dom 𝐹⟶∪ 𝐽) |
60 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:dom 𝐹⟶∪ 𝐽 ∧ 𝑘 ∈ dom 𝐹) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘))) |
61 | 59, 60 | sylan 487 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘))) |
62 | 61 | eleq1d 2672 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → (((𝐺 ∘ 𝐹)‘𝑘) ∈ (𝐺 “ 𝑣) ↔ (𝐺‘(𝐹‘𝑘)) ∈ (𝐺 “ 𝑣))) |
63 | 58, 62 | sylibrd 248 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → ((𝐺 ∘ 𝐹)‘𝑘) ∈ (𝐺 “ 𝑣))) |
64 | | simplrr 797 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → (𝐺 “ 𝑣) ⊆ 𝑢) |
65 | 64 | sseld 3567 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → (((𝐺 ∘ 𝐹)‘𝑘) ∈ (𝐺 “ 𝑣) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)) |
66 | 63, 65 | syld 46 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)) |
67 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑘 ∈ dom 𝐹) |
68 | 26 | ad3antrrr 762 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → dom (𝐺 ∘ 𝐹) = dom 𝐹) |
69 | 67, 68 | eleqtrrd 2691 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → 𝑘 ∈ dom (𝐺 ∘ 𝐹)) |
70 | 66, 69 | jctild 564 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ∈ 𝑣 → (𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
71 | 70 | expimpd 627 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → (𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
72 | 71 | ralimdv 2946 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
73 | 72 | reximdv 2999 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ (𝑣 ∈ 𝐽 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
74 | 73 | expr 641 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ 𝑣 ∈ 𝐽) → ((𝐺 “ 𝑣) ⊆ 𝑢 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))) |
75 | 74 | com23 84 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ 𝑣 ∈ 𝐽) → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) → ((𝐺 “ 𝑣) ⊆ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)))) |
76 | 75 | impd 446 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) ∧ 𝑣 ∈ 𝐽) → ((∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
77 | 76 | rexlimdva 3013 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → (∃𝑣 ∈ 𝐽 (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣) ∧ (𝐺 “ 𝑣) ⊆ 𝑢) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
78 | 49, 77 | syl5 33 |
. . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ((∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐺 “ 𝑣) ⊆ 𝑢)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
79 | 41, 44, 78 | mp2and 711 |
. . . 4
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐾 ∧ (𝐺‘𝑃) ∈ 𝑢)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢)) |
80 | 79 | expr 641 |
. . 3
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐾) → ((𝐺‘𝑃) ∈ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
81 | 80 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑢 ∈ 𝐾 ((𝐺‘𝑃) ∈ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))) |
82 | 3 | toptopon 20548 |
. . . 4
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
83 | 32, 82 | sylib 207 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
84 | 83, 11, 12 | lmbr2 20873 |
. 2
⊢ (𝜑 → ((𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝐾 ↑pm
ℂ) ∧ (𝐺‘𝑃) ∈ ∪ 𝐾 ∧ ∀𝑢 ∈ 𝐾 ((𝐺‘𝑃) ∈ 𝑢 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹)‘𝑘) ∈ 𝑢))))) |
85 | 37, 39, 81, 84 | mpbir3and 1238 |
1
⊢ (𝜑 → (𝐺 ∘ 𝐹)(⇝𝑡‘𝐾)(𝐺‘𝑃)) |