| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | tcvalg 9779 | . . . . 5
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | 
| 2 |  | fvex 6918 | . . . . 5
⊢
(TC‘𝐴) ∈
V | 
| 3 | 1, 2 | eqeltrrdi 2849 | . . . 4
⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) | 
| 4 |  | intexab 5345 | . . . 4
⊢
(∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) | 
| 5 | 3, 4 | sylibr 234 | . . 3
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)) | 
| 6 |  | ssin 4238 | . . . . . . . . 9
⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵)) | 
| 7 | 6 | biimpi 216 | . . . . . . . 8
⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑥 ∩ 𝐵)) | 
| 8 |  | trin 5270 | . . . . . . . 8
⊢ ((Tr
𝑥 ∧ Tr 𝐵) → Tr (𝑥 ∩ 𝐵)) | 
| 9 | 7, 8 | anim12i 613 | . . . . . . 7
⊢ (((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))) | 
| 10 | 9 | an4s 660 | . . . . . 6
⊢ (((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ∧ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))) | 
| 11 | 10 | expcom 413 | . . . . 5
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))) | 
| 12 |  | vex 3483 | . . . . . . . . 9
⊢ 𝑥 ∈ V | 
| 13 | 12 | inex1 5316 | . . . . . . . 8
⊢ (𝑥 ∩ 𝐵) ∈ V | 
| 14 |  | sseq2 4009 | . . . . . . . . 9
⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵))) | 
| 15 |  | treq 5266 | . . . . . . . . 9
⊢ (𝑦 = (𝑥 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝑥 ∩ 𝐵))) | 
| 16 | 14, 15 | anbi12d 632 | . . . . . . . 8
⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))) | 
| 17 | 13, 16 | elab 3678 | . . . . . . 7
⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))) | 
| 18 |  | intss1 4962 | . . . . . . 7
⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵)) | 
| 19 | 17, 18 | sylbir 235 | . . . . . 6
⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩
{𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵)) | 
| 20 |  | inss2 4237 | . . . . . 6
⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵 | 
| 21 | 19, 20 | sstrdi 3995 | . . . . 5
⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩
{𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵) | 
| 22 | 11, 21 | syl6 35 | . . . 4
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) | 
| 23 | 22 | exlimdv 1932 | . . 3
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) | 
| 24 | 5, 23 | syl5com 31 | . 2
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) | 
| 25 |  | tcvalg 9779 | . . 3
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}) | 
| 26 | 25 | sseq1d 4014 | . 2
⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) ⊆ 𝐵 ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) | 
| 27 | 24, 26 | sylibrd 259 | 1
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵)) |