Step | Hyp | Ref
| Expression |
1 | | ptcn.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | 1 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋)) |
3 | | ptcn.5 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐼⟶Top) |
4 | 3 | ffvelrnda 6861 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top) |
5 | | toptopon2 21669 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
6 | 4, 5 | sylib 221 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
7 | | ptcn.6 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) |
8 | | cnf2 22000 |
. . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
9 | 2, 6, 7, 8 | syl3anc 1372 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
10 | 9 | fvmptelrn 6887 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
11 | 10 | an32s 652 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
12 | 11 | ralrimiva 3096 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)) |
13 | | ptcn.4 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
14 | 13 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
15 | | mptelixpg 8545 |
. . . . . 6
⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))) |
16 | 14, 15 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))) |
17 | 12, 16 | mpbird 260 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)) |
18 | | ptcn.2 |
. . . . . . 7
⊢ 𝐾 =
(∏t‘𝐹) |
19 | 18 | ptuni 22345 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
20 | 13, 3, 19 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
21 | 20 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) |
22 | 17, 21 | eleqtrd 2835 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾) |
23 | 22 | fmpttd 6889 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾) |
24 | 1 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
25 | 13 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
26 | 3 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top) |
27 | | simpr 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋) |
28 | 7 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘))) |
29 | | simplr 769 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ 𝑋) |
30 | | toponuni 21665 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
31 | 1, 30 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
32 | 31 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑋 = ∪ 𝐽) |
33 | 29, 32 | eleqtrd 2835 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽) |
34 | | eqid 2738 |
. . . . . 6
⊢ ∪ 𝐽 =
∪ 𝐽 |
35 | 34 | cncnpi 22029 |
. . . . 5
⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) |
36 | 28, 33, 35 | syl2anc 587 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) |
37 | 18, 24, 25, 26, 27, 36 | ptcnp 22373 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) |
38 | 37 | ralrimiva 3096 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) |
39 | | pttop 22333 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) →
(∏t‘𝐹) ∈ Top) |
40 | 13, 3, 39 | syl2anc 587 |
. . . . 5
⊢ (𝜑 →
(∏t‘𝐹) ∈ Top) |
41 | 18, 40 | eqeltrid 2837 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
42 | | toptopon2 21669 |
. . . 4
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
43 | 41, 42 | sylib 221 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
44 | | cncnp 22031 |
. . 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)))) |
45 | 1, 43, 44 | syl2anc 587 |
. 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)))) |
46 | 23, 38, 45 | mpbir2and 713 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾)) |