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Theorem ordtrest 21802
 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 5214 . . . 4 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
21adantr 483 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3 eqid 2819 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4 eqid 2819 . . . 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 2819 . . . 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 21789 . . 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 21800 . . . 4 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9 resttop 21760 . . . 4 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
108, 9sylan 582 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11 eqid 2819 . . . . . . . 8 dom 𝑅 = dom 𝑅
1211psssdm2 17817 . . . . . . 7 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
1312adantr 483 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
148adantr 483 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ Top)
15 simpr 487 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝐴𝑉)
1611ordttopon 21793 . . . . . . . . 9 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
1716adantr 483 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18 toponmax 21526 . . . . . . . 8 ((ordTop‘𝑅) ∈ (TopOn‘dom 𝑅) → dom 𝑅 ∈ (ordTop‘𝑅))
1917, 18syl 17 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom 𝑅 ∈ (ordTop‘𝑅))
20 elrestr 16694 . . . . . . 7 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2114, 15, 19, 20syl3anc 1365 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2213, 21eqeltrd 2911 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2322snssd 4734 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
2413rabeqdv 3483 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
2513, 24mpteq12dv 5142 . . . . . . 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}))
2625rneqd 5801 . . . . . 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}))
27 inrab2 4274 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥}
28 simpr 487 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦 ∈ (dom 𝑅𝐴))
2928elin2d 4174 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦𝐴)
30 simpr 487 . . . . . . . . . . . . . . 15 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥 ∈ (dom 𝑅𝐴))
3130elin2d 4174 . . . . . . . . . . . . . 14 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
3231adantr 483 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
33 brinxp 5623 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3429, 32, 33syl2anc 586 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3534notbid 320 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3635rabbidva 3477 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
3727, 36syl5eq 2866 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
3814adantr 483 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → (ordTop‘𝑅) ∈ Top)
3915adantr 483 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝐴𝑉)
40 simpl 485 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝑅 ∈ PosetRel)
41 elinel1 4170 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ dom 𝑅)
4211ordtopn1 21794 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4340, 41, 42syl2an 597 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
44 elrestr 16694 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4538, 39, 43, 44syl3anc 1365 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4637, 45eqeltrrd 2912 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘𝑅) ↾t 𝐴))
4746fmpttd 6872 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
4847frnd 6514 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
4926, 48eqsstrd 4003 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
5013rabeqdv 3483 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5113, 50mpteq12dv 5142 . . . . . . 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}))
5251rneqd 5801 . . . . . 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}))
53 inrab2 4274 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦}
54 brinxp 5623 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5532, 29, 54syl2anc 586 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5655notbid 320 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5756rabbidva 3477 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5853, 57syl5eq 2866 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5911ordtopn2 21795 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
6040, 41, 59syl2an 597 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
61 elrestr 16694 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6238, 39, 60, 61syl3anc 1365 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6358, 62eqeltrrd 2912 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘𝑅) ↾t 𝐴))
6463fmpttd 6872 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
6564frnd 6514 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
6652, 65eqsstrd 4003 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
6749, 66unssd 4160 . . . 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))
6823, 67unssd 4160 . . 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))
69 tgfiss 21591 . . 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))
7010, 68, 69syl2anc 586 . 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))
717, 70eqsstrd 4003 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 208   ∧ wa 398   = wceq 1530   ∈ wcel 2107  {crab 3140  Vcvv 3493   ∪ cun 3932   ∩ cin 3933   ⊆ wss 3934  {csn 4559   class class class wbr 5057   ↦ cmpt 5137   × cxp 5546  dom cdm 5548  ran crn 5549  ‘cfv 6348  (class class class)co 7148  ficfi 8866   ↾t crest 16686  topGenctg 16703  ordTopcordt 16764  PosetRelcps 17800  Topctop 21493  TopOnctopon 21510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-fin 8505  df-fi 8867  df-rest 16688  df-topgen 16709  df-ordt 16766  df-ps 17802  df-top 21494  df-topon 21511  df-bases 21546 This theorem is referenced by:  ordtrest2  21804
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