Step | Hyp | Ref
| Expression |
1 | | ptcn.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | 1 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋)) |
3 | | ptcn.5 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:𝐼⟶Top) |
4 | 3 | ffvelrnda 6608 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top) |
5 | | eqid 2825 |
. . . . . . . . . . . 12
⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) |
6 | 5 | toptopon 21092 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
7 | 4, 6 | sylib 210 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
8 | | ptcn.6 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) |
9 | | cnf2 21424 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
10 | 2, 7, 8, 9 | syl3anc 1496 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
11 | | eqid 2825 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
12 | 11 | fmpt 6629 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝑋 𝐴 ∈ ∪ (𝐹‘𝑘) ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
13 | 10, 12 | sylibr 226 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘)) |
14 | 13 | r19.21bi 3141 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
15 | 14 | an32s 644 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
16 | 15 | ralrimiva 3175 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)) |
17 | | ptcn.4 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
18 | 17 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
19 | | mptelixpg 8212 |
. . . . . 6
⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))) |
20 | 18, 19 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))) |
21 | 16, 20 | mpbird 249 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)) |
22 | | ptcn.2 |
. . . . . . 7
⊢ 𝐾 =
(∏t‘𝐹) |
23 | 22 | ptuni 21768 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
24 | 17, 3, 23 | syl2anc 581 |
. . . . 5
⊢ (𝜑 → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
25 | 24 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
26 | 21, 25 | eleqtrd 2908 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾) |
27 | 26 | fmpttd 6634 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾) |
28 | 1 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
29 | 17 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
30 | 3 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top) |
31 | | simpr 479 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋) |
32 | 8 | adantlr 708 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) |
33 | | simplr 787 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ 𝑋) |
34 | | toponuni 21089 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
35 | 1, 34 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
36 | 35 | ad2antrr 719 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑋 = ∪ 𝐽) |
37 | 33, 36 | eleqtrd 2908 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽) |
38 | | eqid 2825 |
. . . . . 6
⊢ ∪ 𝐽 =
∪ 𝐽 |
39 | 38 | cncnpi 21453 |
. . . . 5
⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) |
40 | 32, 37, 39 | syl2anc 581 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) |
41 | 22, 28, 29, 30, 31, 40 | ptcnp 21796 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) |
42 | 41 | ralrimiva 3175 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) |
43 | | pttop 21756 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) →
(∏t‘𝐹) ∈ Top) |
44 | 17, 3, 43 | syl2anc 581 |
. . . . 5
⊢ (𝜑 →
(∏t‘𝐹) ∈ Top) |
45 | 22, 44 | syl5eqel 2910 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
46 | | eqid 2825 |
. . . . 5
⊢ ∪ 𝐾 =
∪ 𝐾 |
47 | 46 | toptopon 21092 |
. . . 4
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
48 | 45, 47 | sylib 210 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
49 | | cncnp 21455 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾))
→ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾 ∧ ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)))) |
50 | 1, 48, 49 | syl2anc 581 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾 ∧ ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)))) |
51 | 27, 42, 50 | mpbir2and 706 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾)) |