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Theorem ordtrest 23211
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 5318 . . . 4 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
21adantr 480 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (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})
63, 4, 5ordtval 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}))))))
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 23209 . . . 4 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9 resttop 23169 . . . 4 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
108, 9sylan 580 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11 eqid 2736 . . . . . . . 8 dom 𝑅 = dom 𝑅
1211psssdm2 18627 . . . . . . 7 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
1312adantr 480 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
148adantr 480 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ Top)
15 simpr 484 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝐴𝑉)
1611ordttopon 23202 . . . . . . . . 9 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
1716adantr 480 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18 toponmax 22933 . . . . . . . 8 ((ordTop‘𝑅) ∈ (TopOn‘dom 𝑅) → dom 𝑅 ∈ (ordTop‘𝑅))
1917, 18syl 17 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom 𝑅 ∈ (ordTop‘𝑅))
20 elrestr 17474 . . . . . . 7 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2114, 15, 19, 20syl3anc 1372 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2213, 21eqeltrd 2840 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2322snssd 4808 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
2413rabeqdv 3451 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
2513, 24mpteq12dv 5232 . . . . . . 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 5948 . . . . . 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 4316 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥}
28 simpr 484 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦 ∈ (dom 𝑅𝐴))
2928elin2d 4204 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦𝐴)
30 simpr 484 . . . . . . . . . . . . . . 15 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥 ∈ (dom 𝑅𝐴))
3130elin2d 4204 . . . . . . . . . . . . . 14 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
3231adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
33 brinxp 5763 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3429, 32, 33syl2anc 584 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3534notbid 318 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3635rabbidva 3442 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
3727, 36eqtrid 2788 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
3814adantr 480 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → (ordTop‘𝑅) ∈ Top)
3915adantr 480 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝐴𝑉)
40 simpl 482 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝑅 ∈ PosetRel)
41 elinel1 4200 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ dom 𝑅)
4211ordtopn1 23203 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4340, 41, 42syl2an 596 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
44 elrestr 17474 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4538, 39, 43, 44syl3anc 1372 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4637, 45eqeltrrd 2841 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘𝑅) ↾t 𝐴))
4746fmpttd 7134 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
4847frnd 6743 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
4926, 48eqsstrd 4017 . . . . 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 3451 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5113, 50mpteq12dv 5232 . . . . . . 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 5948 . . . . . 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 4316 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦}
54 brinxp 5763 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5532, 29, 54syl2anc 584 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5655notbid 318 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5756rabbidva 3442 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5853, 57eqtrid 2788 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5911ordtopn2 23204 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
6040, 41, 59syl2an 596 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
61 elrestr 17474 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6238, 39, 60, 61syl3anc 1372 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6358, 62eqeltrrd 2841 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘𝑅) ↾t 𝐴))
6463fmpttd 7134 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
6564frnd 6743 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
6652, 65eqsstrd 4017 . . . . 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 4191 . . . 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 4191 . . 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 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))
7010, 68, 69syl2anc 584 . 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 4017 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 206  wa 395   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  cun 3948  cin 3949  wss 3950  {csn 4625   class class class wbr 5142  cmpt 5224   × cxp 5682  dom cdm 5684  ran crn 5685  cfv 6560  (class class class)co 7432  ficfi 9451  t crest 17466  topGenctg 17483  ordTopcordt 17545  PosetRelcps 18610  Topctop 22900  TopOnctopon 22917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-2o 8508  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-ordt 17547  df-ps 18612  df-top 22901  df-topon 22918  df-bases 22954
This theorem is referenced by:  ordtrest2  23213
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