| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | inex1g 5318 | . . . 4
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V) | 
| 2 | 1 | adantr 480 | . . 3
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V) | 
| 3 |  | eqid 2736 | . . . 4
⊢ dom
(𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴)) | 
| 4 |  | eqid 2736 | . . . 4
⊢ ran
(𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) | 
| 5 |  | eqid 2736 | . . . 4
⊢ ran
(𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) | 
| 6 | 3, 4, 5 | ordtval 23198 | . . 3
⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (topGen‘(fi‘({dom (𝑅 ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})))))) | 
| 7 | 2, 6 | syl 17 | . 2
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (topGen‘(fi‘({dom (𝑅 ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})))))) | 
| 8 |  | ordttop 23209 | . . . 4
⊢ (𝑅 ∈ PosetRel →
(ordTop‘𝑅) ∈
Top) | 
| 9 |  | resttop 23169 | . . . 4
⊢
(((ordTop‘𝑅)
∈ Top ∧ 𝐴 ∈
𝑉) →
((ordTop‘𝑅)
↾t 𝐴)
∈ Top) | 
| 10 | 8, 9 | sylan 580 | . . 3
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top) | 
| 11 |  | eqid 2736 | . . . . . . . 8
⊢ dom 𝑅 = dom 𝑅 | 
| 12 | 11 | psssdm2 18627 | . . . . . . 7
⊢ (𝑅 ∈ PosetRel → dom
(𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅 ∩ 𝐴)) | 
| 13 | 12 | adantr 480 | . . . . . 6
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅 ∩ 𝐴)) | 
| 14 | 8 | adantr 480 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ Top) | 
| 15 |  | simpr 484 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | 
| 16 | 11 | ordttopon 23202 | . . . . . . . . 9
⊢ (𝑅 ∈ PosetRel →
(ordTop‘𝑅) ∈
(TopOn‘dom 𝑅)) | 
| 17 | 16 | adantr 480 | . . . . . . . 8
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅)) | 
| 18 |  | toponmax 22933 | . . . . . . . 8
⊢
((ordTop‘𝑅)
∈ (TopOn‘dom 𝑅)
→ dom 𝑅 ∈
(ordTop‘𝑅)) | 
| 19 | 17, 18 | syl 17 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom 𝑅 ∈ (ordTop‘𝑅)) | 
| 20 |  | elrestr 17474 | . . . . . . 7
⊢
(((ordTop‘𝑅)
∈ Top ∧ 𝐴 ∈
𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 21 | 14, 15, 19, 20 | syl3anc 1372 | . . . . . 6
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 22 | 13, 21 | eqeltrd 2840 | . . . . 5
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 23 | 22 | snssd 4808 | . . . 4
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 24 | 13 | rabeqdv 3451 | . . . . . . . 8
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) | 
| 25 | 13, 24 | mpteq12dv 5232 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})) | 
| 26 | 25 | rneqd 5948 | . . . . . 6
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})) | 
| 27 |  | inrab2 4316 | . . . . . . . . . 10
⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦𝑅𝑥} | 
| 28 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) | 
| 29 | 28 | elin2d 4204 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑦 ∈ 𝐴) | 
| 30 |  | simpr 484 | . . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) | 
| 31 | 30 | elin2d 4204 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑥 ∈ 𝐴) | 
| 32 | 31 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑥 ∈ 𝐴) | 
| 33 |  | brinxp 5763 | . . . . . . . . . . . . 13
⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 34 | 29, 32, 33 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 35 | 34 | notbid 318 | . . . . . . . . . . 11
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 36 | 35 | rabbidva 3442 | . . . . . . . . . 10
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) | 
| 37 | 27, 36 | eqtrid 2788 | . . . . . . . . 9
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) | 
| 38 | 14 | adantr 480 | . . . . . . . . . 10
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → (ordTop‘𝑅) ∈ Top) | 
| 39 | 15 | adantr 480 | . . . . . . . . . 10
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → 𝐴 ∈ 𝑉) | 
| 40 |  | simpl 482 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ PosetRel) | 
| 41 |  | elinel1 4200 | . . . . . . . . . . 11
⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ dom 𝑅) | 
| 42 | 11 | ordtopn1 23203 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) | 
| 43 | 40, 41, 42 | syl2an 596 | . . . . . . . . . 10
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) | 
| 44 |  | elrestr 17474 | . . . . . . . . . 10
⊢
(((ordTop‘𝑅)
∈ Top ∧ 𝐴 ∈
𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 45 | 38, 39, 43, 44 | syl3anc 1372 | . . . . . . . . 9
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 46 | 37, 45 | eqeltrrd 2841 | . . . . . . . 8
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 47 | 46 | fmpttd 7134 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}):(dom 𝑅 ∩ 𝐴)⟶((ordTop‘𝑅) ↾t 𝐴)) | 
| 48 | 47 | frnd 6743 | . . . . . 6
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ran (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 49 | 26, 48 | eqsstrd 4017 | . . . . 5
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 50 | 13 | rabeqdv 3451 | . . . . . . . 8
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) | 
| 51 | 13, 50 | mpteq12dv 5232 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})) | 
| 52 | 51 | rneqd 5948 | . . . . . 6
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})) | 
| 53 |  | inrab2 4316 | . . . . . . . . . 10
⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥𝑅𝑦} | 
| 54 |  | brinxp 5763 | . . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦)) | 
| 55 | 32, 29, 54 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦)) | 
| 56 | 55 | notbid 318 | . . . . . . . . . . 11
⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦)) | 
| 57 | 56 | rabbidva 3442 | . . . . . . . . . 10
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) | 
| 58 | 53, 57 | eqtrid 2788 | . . . . . . . . 9
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) | 
| 59 | 11 | ordtopn2 23204 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) | 
| 60 | 40, 41, 59 | syl2an 596 | . . . . . . . . . 10
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) | 
| 61 |  | elrestr 17474 | . . . . . . . . . 10
⊢
(((ordTop‘𝑅)
∈ Top ∧ 𝐴 ∈
𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 62 | 38, 39, 60, 61 | syl3anc 1372 | . . . . . . . . 9
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 63 | 58, 62 | eqeltrrd 2841 | . . . . . . . 8
⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 64 | 63 | fmpttd 7134 | . . . . . . 7
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}):(dom 𝑅 ∩ 𝐴)⟶((ordTop‘𝑅) ↾t 𝐴)) | 
| 65 | 64 | frnd 6743 | . . . . . 6
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ran (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↦ {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 66 | 52, 65 | eqsstrd 4017 | . . . . 5
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 67 | 49, 66 | unssd 4191 | . . . 4
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 68 | 23, 67 | unssd 4191 | . . 3
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ({dom (𝑅 ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}))) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 69 |  | tgfiss 22999 | . . 3
⊢
((((ordTop‘𝑅)
↾t 𝐴)
∈ Top ∧ ({dom (𝑅
∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}))) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) → (topGen‘(fi‘({dom
(𝑅 ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}))))) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 70 | 10, 68, 69 | syl2anc 584 | . 2
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (topGen‘(fi‘({dom
(𝑅 ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}))))) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | 
| 71 | 7, 70 | eqsstrd 4017 | 1
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) |