Step | Hyp | Ref
| Expression |
1 | | ptcn.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | 1 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋)) |
3 | | ptcn.5 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:𝐼⟶Top) |
4 | 3 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top) |
5 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) |
6 | 5 | toptopon 20548 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
7 | 4, 6 | sylib 207 |
. . . . . . . . . 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 20863 |
. . . . . . . . . 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 1318 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
11 | | eqid 2610 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
12 | 11 | fmpt 6289 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝑋 𝐴 ∈ ∪ (𝐹‘𝑘) ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
13 | 10, 12 | sylibr 223 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘)) |
14 | 13 | r19.21bi 2916 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
15 | 14 | an32s 842 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
16 | 15 | ralrimiva 2949 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)) |
17 | | ptcn.4 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
18 | 17 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
19 | | mptelixpg 7831 |
. . . . . 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 246 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)) |
22 | | ptcn.2 |
. . . . . . 7
⊢ 𝐾 =
(∏t‘𝐹) |
23 | 22 | ptuni 21207 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
24 | 17, 3, 23 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
25 | 24 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
26 | 21, 25 | eleqtrd 2690 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾) |
27 | | eqid 2610 |
. . 3
⊢ (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) |
28 | 26, 27 | fmptd 6292 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾) |
29 | 1 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
30 | 17 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
31 | 3 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top) |
32 | | simpr 476 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋) |
33 | 8 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) |
34 | | simplr 788 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ 𝑋) |
35 | | toponuni 20542 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
36 | 1, 35 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
37 | 36 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑋 = ∪ 𝐽) |
38 | 34, 37 | eleqtrd 2690 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽) |
39 | | eqid 2610 |
. . . . . 6
⊢ ∪ 𝐽 =
∪ 𝐽 |
40 | 39 | cncnpi 20892 |
. . . . 5
⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) |
41 | 33, 38, 40 | syl2anc 691 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) |
42 | 22, 29, 30, 31, 32, 41 | ptcnp 21235 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) |
43 | 42 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) |
44 | | pttop 21195 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) →
(∏t‘𝐹) ∈ Top) |
45 | 17, 3, 44 | syl2anc 691 |
. . . . 5
⊢ (𝜑 →
(∏t‘𝐹) ∈ Top) |
46 | 22, 45 | syl5eqel 2692 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
47 | | eqid 2610 |
. . . . 5
⊢ ∪ 𝐾 =
∪ 𝐾 |
48 | 47 | toptopon 20548 |
. . . 4
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
49 | 46, 48 | sylib 207 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
50 | | cncnp 20894 |
. . 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)))) |
51 | 1, 49, 50 | syl2anc 691 |
. 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)))) |
52 | 28, 43, 51 | mpbir2and 959 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾)) |