| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | psss 18625 | . . 3
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel) | 
| 2 |  | inss1 4237 | . . . . . 6
⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 | 
| 3 |  | dmss 5913 | . . . . . 6
⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅) | 
| 4 |  | ssralv 4052 | . . . . . 6
⊢ (dom
(𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅 → (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) | 
| 5 | 2, 3, 4 | mp2b 10 | . . . . 5
⊢
(∀𝑥 ∈
dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) | 
| 6 |  | ssralv 4052 | . . . . . . 7
⊢ (dom
(𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅 → (∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) | 
| 7 | 2, 3, 6 | mp2b 10 | . . . . . 6
⊢
(∀𝑦 ∈
dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) | 
| 8 | 7 | ralimi 3083 | . . . . 5
⊢
(∀𝑥 ∈
dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) | 
| 9 | 5, 8 | syl 17 | . . . 4
⊢
(∀𝑥 ∈
dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) | 
| 10 |  | inss2 4238 | . . . . . . . . . 10
⊢ (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | 
| 11 |  | dmss 5913 | . . . . . . . . . 10
⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom (𝐴 × 𝐴)) | 
| 12 | 10, 11 | ax-mp 5 | . . . . . . . . 9
⊢ dom
(𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom (𝐴 × 𝐴) | 
| 13 |  | dmxpid 5941 | . . . . . . . . 9
⊢ dom
(𝐴 × 𝐴) = 𝐴 | 
| 14 | 12, 13 | sseqtri 4032 | . . . . . . . 8
⊢ dom
(𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴 | 
| 15 | 14 | sseli 3979 | . . . . . . 7
⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥 ∈ 𝐴) | 
| 16 | 14 | sseli 3979 | . . . . . . 7
⊢ (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦 ∈ 𝐴) | 
| 17 |  | brinxp 5764 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦)) | 
| 18 |  | brinxp 5764 | . . . . . . . . 9
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 19 | 18 | ancoms 458 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 20 | 17, 19 | orbi12d 919 | . . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 21 | 15, 16, 20 | syl2an 596 | . . . . . 6
⊢ ((𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∧ 𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 22 | 21 | ralbidva 3176 | . . . . 5
⊢ (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → (∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 23 | 22 | ralbiia 3091 | . . . 4
⊢
(∀𝑥 ∈
dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 24 | 9, 23 | sylib 218 | . . 3
⊢
(∀𝑥 ∈
dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 25 | 1, 24 | anim12i 613 | . 2
⊢ ((𝑅 ∈ PosetRel ∧
∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ∧ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 26 |  | eqid 2737 | . . 3
⊢ dom 𝑅 = dom 𝑅 | 
| 27 | 26 | istsr2 18629 | . 2
⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧
∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) | 
| 28 |  | eqid 2737 | . . 3
⊢ dom
(𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴)) | 
| 29 | 28 | istsr2 18629 | . 2
⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel ↔ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ∧ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 30 | 25, 27, 29 | 3imtr4i 292 | 1
⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel ) |