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Theorem ordtrest 21200
Description: The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
ordtrest ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))

Proof of Theorem ordtrest
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inex1g 4945 . . . 4 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
21adantr 472 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3 eqid 2752 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4 eqid 2752 . . . 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 2752 . . . 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})
63, 4, 5ordtval 21187 . . 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}))))))
72, 6syl 17 . 2 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (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 21198 . . . 4 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9 resttop 21158 . . . 4 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
108, 9sylan 489 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11 eqid 2752 . . . . . . . 8 dom 𝑅 = dom 𝑅
1211psssdm2 17408 . . . . . . 7 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
1312adantr 472 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
148adantr 472 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ Top)
15 simpr 479 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝐴𝑉)
1611ordttopon 21191 . . . . . . . . 9 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
1716adantr 472 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18 toponmax 20924 . . . . . . . 8 ((ordTop‘𝑅) ∈ (TopOn‘dom 𝑅) → dom 𝑅 ∈ (ordTop‘𝑅))
1917, 18syl 17 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom 𝑅 ∈ (ordTop‘𝑅))
20 elrestr 16283 . . . . . . 7 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2114, 15, 19, 20syl3anc 1473 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2213, 21eqeltrd 2831 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2322snssd 4477 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
24 rabeq 3324 . . . . . . . . 9 (dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
2513, 24syl 17 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
2613, 25mpteq12dv 4877 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}))
2726rneqd 5500 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}))
28 inrab2 4035 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥}
29 inss2 3969 . . . . . . . . . . . . . 14 (dom 𝑅𝐴) ⊆ 𝐴
30 simpr 479 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦 ∈ (dom 𝑅𝐴))
3129, 30sseldi 3734 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦𝐴)
32 simpr 479 . . . . . . . . . . . . . . 15 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥 ∈ (dom 𝑅𝐴))
3329, 32sseldi 3734 . . . . . . . . . . . . . 14 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
3433adantr 472 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
35 brinxp 5330 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3631, 34, 35syl2anc 696 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3736notbid 307 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3837rabbidva 3320 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
3928, 38syl5eq 2798 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
4014adantr 472 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → (ordTop‘𝑅) ∈ Top)
4115adantr 472 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝐴𝑉)
42 simpl 474 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝑅 ∈ PosetRel)
43 inss1 3968 . . . . . . . . . . . 12 (dom 𝑅𝐴) ⊆ dom 𝑅
4443sseli 3732 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ dom 𝑅)
4511ordtopn1 21192 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4642, 44, 45syl2an 495 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
47 elrestr 16283 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4840, 41, 46, 47syl3anc 1473 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4939, 48eqeltrrd 2832 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘𝑅) ↾t 𝐴))
50 eqid 2752 . . . . . . . 8 (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
5149, 50fmptd 6540 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
52 frn 6206 . . . . . . 7 ((𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
5351, 52syl 17 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
5427, 53eqsstrd 3772 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
55 rabeq 3324 . . . . . . . . 9 (dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5613, 55syl 17 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5713, 56mpteq12dv 4877 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}))
5857rneqd 5500 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}))
59 inrab2 4035 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦}
60 brinxp 5330 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6134, 31, 60syl2anc 696 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6261notbid 307 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6362rabbidva 3320 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
6459, 63syl5eq 2798 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
6511ordtopn2 21193 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
6642, 44, 65syl2an 495 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
67 elrestr 16283 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6840, 41, 66, 67syl3anc 1473 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6964, 68eqeltrrd 2832 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘𝑅) ↾t 𝐴))
70 eqid 2752 . . . . . . . 8 (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
7169, 70fmptd 6540 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
72 frn 6206 . . . . . . 7 ((𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
7371, 72syl 17 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
7458, 73eqsstrd 3772 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
7554, 74unssd 3924 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (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))
7623, 75unssd 3924 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ({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))
77 tgfiss 20989 . . 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))
7810, 76, 77syl2anc 696 . 2 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (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))
797, 78eqsstrd 3772 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  {crab 3046  Vcvv 3332  cun 3705  cin 3706  wss 3707  {csn 4313   class class class wbr 4796  cmpt 4873   × cxp 5256  dom cdm 5258  ran crn 5259  wf 6037  cfv 6041  (class class class)co 6805  ficfi 8473  t crest 16275  topGenctg 16292  ordTopcordt 16353  PosetRelcps 17391  Topctop 20892  TopOnctopon 20909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-en 8114  df-fin 8117  df-fi 8474  df-rest 16277  df-topgen 16298  df-ordt 16355  df-ps 17393  df-top 20893  df-topon 20910  df-bases 20944
This theorem is referenced by:  ordtrest2  21202
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