| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ptcn.3 | . . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | 
| 2 | 1 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 3 |  | ptcn.5 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐼⟶Top) | 
| 4 | 3 | ffvelcdmda 7103 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top) | 
| 5 |  | toptopon2 22925 | . . . . . . . . . 10
⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) | 
| 6 | 4, 5 | sylib 218 | . . . . . . . . 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 23258 | . . . . . . . . 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 | fvmptelcdm 7132 | . . . . . . 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 3145 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)) | 
| 13 |  | ptcn.4 | . . . . . . 7
⊢ (𝜑 → 𝐼 ∈ 𝑉) | 
| 14 | 13 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉) | 
| 15 |  | mptelixpg 8976 | . . . . . 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 257 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)) | 
| 18 |  | ptcn.2 | . . . . . . 7
⊢ 𝐾 =
(∏t‘𝐹) | 
| 19 | 18 | ptuni 23603 | . . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) | 
| 20 | 13, 3, 19 | syl2anc 584 | . . . . 5
⊢ (𝜑 → X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) | 
| 21 | 20 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) = ∪ 𝐾) | 
| 22 | 17, 21 | eleqtrd 2842 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ ∪ 𝐾) | 
| 23 | 22 | fmpttd 7134 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶∪ 𝐾) | 
| 24 | 1 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 25 | 13 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐼 ∈ 𝑉) | 
| 26 | 3 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐹:𝐼⟶Top) | 
| 27 |  | simpr 484 | . . . 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 768 | . . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ 𝑋) | 
| 30 |  | toponuni 22921 | . . . . . . . 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 2842 | . . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑧 ∈ ∪ 𝐽) | 
| 34 |  | eqid 2736 | . . . . . 6
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 35 | 34 | cncnpi 23287 | . . . . 5
⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)) ∧ 𝑧 ∈ ∪ 𝐽) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) | 
| 36 | 28, 33, 35 | syl2anc 584 | . . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝑧)) | 
| 37 | 18, 24, 25, 26, 27, 36 | ptcnp 23631 | . . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) | 
| 38 | 37 | ralrimiva 3145 | . 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝑧)) | 
| 39 |  | pttop 23591 | . . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶Top) →
(∏t‘𝐹) ∈ Top) | 
| 40 | 13, 3, 39 | syl2anc 584 | . . . . 5
⊢ (𝜑 →
(∏t‘𝐹) ∈ Top) | 
| 41 | 18, 40 | eqeltrid 2844 | . . . 4
⊢ (𝜑 → 𝐾 ∈ Top) | 
| 42 |  | toptopon2 22925 | . . . 4
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) | 
| 43 | 41, 42 | sylib 218 | . . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾)) | 
| 44 |  | cncnp 23289 | . . 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 584 | . 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)) |