| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elfvunirn 6938 | . . . . 5
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran
UnifOn) | 
| 2 |  | unieq 4918 | . . . . . . . . 9
⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪
𝑈) | 
| 3 | 2 | dmeqd 5916 | . . . . . . . 8
⊢ (𝑢 = 𝑈 → dom ∪
𝑢 = dom ∪ 𝑈) | 
| 4 | 3 | fveq2d 6910 | . . . . . . 7
⊢ (𝑢 = 𝑈 → (fBas‘dom ∪ 𝑢) =
(fBas‘dom ∪ 𝑈)) | 
| 5 |  | raleq 3323 | . . . . . . 7
⊢ (𝑢 = 𝑈 → (∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣)) | 
| 6 | 4, 5 | rabeqbidv 3455 | . . . . . 6
⊢ (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom ∪ 𝑢)
∣ ∀𝑣 ∈
𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) | 
| 7 |  | df-cfilu 24296 | . . . . . 6
⊢
CauFilu = (𝑢 ∈ ∪ ran
UnifOn ↦ {𝑓 ∈
(fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) | 
| 8 |  | fvex 6919 | . . . . . . 7
⊢
(fBas‘dom ∪ 𝑈) ∈ V | 
| 9 | 8 | rabex 5339 | . . . . . 6
⊢ {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V | 
| 10 | 6, 7, 9 | fvmpt 7016 | . . . . 5
⊢ (𝑈 ∈ ∪ ran UnifOn → (CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) | 
| 11 | 1, 10 | syl 17 | . . . 4
⊢ (𝑈 ∈ (UnifOn‘𝑋) →
(CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) | 
| 12 | 11 | eleq2d 2827 | . . 3
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})) | 
| 13 |  | rexeq 3322 | . . . . 5
⊢ (𝑓 = 𝐹 → (∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) | 
| 14 | 13 | ralbidv 3178 | . . . 4
⊢ (𝑓 = 𝐹 → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) | 
| 15 | 14 | elrab 3692 | . . 3
⊢ (𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈)
∧ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) | 
| 16 | 12, 15 | bitrdi 287 | . 2
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈)
∧ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) | 
| 17 |  | ustbas2 24234 | . . . . 5
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈) | 
| 18 | 17 | fveq2d 6910 | . . . 4
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom ∪ 𝑈)) | 
| 19 | 18 | eleq2d 2827 | . . 3
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom ∪ 𝑈))) | 
| 20 | 19 | anbi1d 631 | . 2
⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈)
∧ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) | 
| 21 | 16, 20 | bitr4d 282 | 1
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) |