Theorem ordtrest 23231

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 ∧ 𝐴 ∈ 𝑉) → (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.

Step	Hyp	Ref	Expression
1		inex1g 5337	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3		eqid 2740	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4		eqid 2740	. . . 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 2740	. . . 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 23218	. . 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 23229	. . . 4 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9		resttop 23189	. . . 4 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
10	8, 9	sylan 579	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11		eqid 2740	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
12	11	psssdm2 18651	. . . . . . 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 23222	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18		toponmax 22953	. . . . . . . 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 17488	. . . . . . 7 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
21	14, 15, 19, 20	syl3anc 1371	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
22	13, 21	eqeltrd 2844	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
23	22	snssd 4834	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
24	13	rabeqdv 3459	. . . . . . . 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 5257	. . . . . . 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 5963	. . . . . 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 4336	. . . . . . . . . 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 4228	. . . . . . . . . . . . 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 4228	. . . . . . . . . . . . . 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 5778	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
34	29, 32, 33	syl2anc 583	. . . . . . . . . . . 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 3450	. . . . . . . . . 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 2792	. . . . . . . . 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 4224	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ dom 𝑅)
42	11	ordtopn1 23223	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
43	40, 41, 42	syl2an 595	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
44		elrestr 17488	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
46	37, 45	eqeltrrd 2845	. . . . . . . 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 7149	. . . . . . 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 6755	. . . . . 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 4047	. . . . 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 3459	. . . . . . . 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 5257	. . . . . . 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 5963	. . . . . 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 4336	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥𝑅𝑦}
54		brinxp 5778	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
55	32, 29, 54	syl2anc 583	. . . . . . . . . . . 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 3450	. . . . . . . . . 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 2792	. . . . . . . . 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 23224	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
60	40, 41, 59	syl2an 595	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
61		elrestr 17488	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
63	58, 62	eqeltrrd 2845	. . . . . . . 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 7149	. . . . . . 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 6755	. . . . . 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 4047	. . . . 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 4215	. . . 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 4215	. . 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 23019	. . 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 583	. 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 4047	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  {csn 4648   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  dom cdm 5700  ran crn 5701  ‘cfv 6573  (class class class)co 7448  ficfi 9479   ↾_t crest 17480  topGenctg 17497  ordTopcordt 17559  PosetRelcps 18634  Topctop 22920  TopOnctopon 22937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-en 9004  df-fin 9007  df-fi 9480  df-rest 17482  df-topgen 17503  df-ordt 17561  df-ps 18636  df-top 22921  df-topon 22938  df-bases 22974

This theorem is referenced by:  ordtrest2  23233