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Theorem ordtrest 21335
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 4996 . . . 4 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
21adantr 473 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3 eqid 2799 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4 eqid 2799 . . . 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 2799 . . . 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 21322 . . 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 21333 . . . 4 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9 resttop 21293 . . . 4 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
108, 9sylan 576 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11 eqid 2799 . . . . . . . 8 dom 𝑅 = dom 𝑅
1211psssdm2 17530 . . . . . . 7 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
1312adantr 473 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅𝐴))
148adantr 473 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ Top)
15 simpr 478 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝐴𝑉)
1611ordttopon 21326 . . . . . . . . 9 (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
1716adantr 473 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18 toponmax 21059 . . . . . . . 8 ((ordTop‘𝑅) ∈ (TopOn‘dom 𝑅) → dom 𝑅 ∈ (ordTop‘𝑅))
1917, 18syl 17 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom 𝑅 ∈ (ordTop‘𝑅))
20 elrestr 16404 . . . . . . 7 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2114, 15, 19, 20syl3anc 1491 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (dom 𝑅𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2213, 21eqeltrd 2878 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
2322snssd 4528 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
24 rabeq 3376 . . . . . . . . 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 4926 . . . . . . 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 5556 . . . . . 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 4100 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥}
29 inss2 4029 . . . . . . . . . . . . . 14 (dom 𝑅𝐴) ⊆ 𝐴
30 simpr 478 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦 ∈ (dom 𝑅𝐴))
3129, 30sseldi 3796 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑦𝐴)
32 simpr 478 . . . . . . . . . . . . . . 15 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥 ∈ (dom 𝑅𝐴))
3329, 32sseldi 3796 . . . . . . . . . . . . . 14 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
3433adantr 473 . . . . . . . . . . . . 13 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → 𝑥𝐴)
35 brinxp 5385 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3631, 34, 35syl2anc 580 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3736notbid 310 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3837rabbidva 3372 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
3928, 38syl5eq 2845 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
4014adantr 473 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → (ordTop‘𝑅) ∈ Top)
4115adantr 473 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → 𝐴𝑉)
42 simpl 475 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → 𝑅 ∈ PosetRel)
43 inss1 4028 . . . . . . . . . . . 12 (dom 𝑅𝐴) ⊆ dom 𝑅
4443sseli 3794 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝑅𝐴) → 𝑥 ∈ dom 𝑅)
4511ordtopn1 21327 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
4642, 44, 45syl2an 590 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
47 elrestr 16404 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4840, 41, 46, 47syl3anc 1491 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
4939, 48eqeltrrd 2879 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘𝑅) ↾t 𝐴))
5049fmpttd 6611 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
5150frnd 6263 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
5227, 51eqsstrd 3835 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
53 rabeq 3376 . . . . . . . . 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})
5413, 53syl 17 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
5513, 54mpteq12dv 4926 . . . . . . 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}))
5655rneqd 5556 . . . . . 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}))
57 inrab2 4100 . . . . . . . . . 10 ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦}
58 brinxp 5385 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
5934, 31, 58syl2anc 580 . . . . . . . . . . . 12 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6059notbid 310 . . . . . . . . . . 11 ((((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) ∧ 𝑦 ∈ (dom 𝑅𝐴)) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6160rabbidva 3372 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
6257, 61syl5eq 2845 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦})
6311ordtopn2 21328 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
6442, 44, 63syl2an 590 . . . . . . . . . 10 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
65 elrestr 16404 . . . . . . . . . 10 (((ordTop‘𝑅) ∈ Top ∧ 𝐴𝑉 ∧ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6640, 41, 64, 65syl3anc 1491 . . . . . . . . 9 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
6762, 66eqeltrrd 2879 . . . . . . . 8 (((𝑅 ∈ PosetRel ∧ 𝐴𝑉) ∧ 𝑥 ∈ (dom 𝑅𝐴)) → {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘𝑅) ↾t 𝐴))
6867fmpttd 6611 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}):(dom 𝑅𝐴)⟶((ordTop‘𝑅) ↾t 𝐴))
6968frnd 6263 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ (dom 𝑅𝐴) ↦ {𝑦 ∈ (dom 𝑅𝐴) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
7056, 69eqsstrd 3835 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → ran (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
7152, 70unssd 3987 . . . 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))
7223, 71unssd 3987 . . 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))
73 tgfiss 21124 . . 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))
7410, 72, 73syl2anc 580 . 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))
757, 74eqsstrd 3835 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 198  wa 385   = wceq 1653  wcel 2157  {crab 3093  Vcvv 3385  cun 3767  cin 3768  wss 3769  {csn 4368   class class class wbr 4843  cmpt 4922   × cxp 5310  dom cdm 5312  ran crn 5313  cfv 6101  (class class class)co 6878  ficfi 8558  t crest 16396  topGenctg 16413  ordTopcordt 16474  PosetRelcps 17513  Topctop 21026  TopOnctopon 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-fin 8199  df-fi 8559  df-rest 16398  df-topgen 16419  df-ordt 16476  df-ps 17515  df-top 21027  df-topon 21044  df-bases 21079
This theorem is referenced by:  ordtrest2  21337
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