Theorem ordtrest 23553

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 5284	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
2	1	adantr 486	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3		eqid 2763	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4		eqid 2763	. . . 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 2763	. . . 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 23540	. . 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 18	. 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 23551	. . . 4 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9		resttop 23511	. . . 4 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
10	8, 9	sylan 592	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11		eqid 2763	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
12	11	psssdm2 18787	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅 ∩ 𝐴))
13	12	adantr 486	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) = (dom 𝑅 ∩ 𝐴))
14	8	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ Top)
15		simpr 490	. . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
16	11	ordttopon 23544	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
17	16	adantr 486	. . . . . . . 8 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18		toponmax 23277	. . . . . . . 8 ⊢ ((ordTop‘𝑅) ∈ (TopOn‘dom 𝑅) → dom 𝑅 ∈ (ordTop‘𝑅))
19	17, 18	syl 18	. . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom 𝑅 ∈ (ordTop‘𝑅))
20		elrestr 17631	. . . . . . 7 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
21	14, 15, 19, 20	syl3anc 1399	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
22	13, 21	eqeltrd 2864	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
23	22	snssd 4747	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
24	13	rabeqdv 3431	. . . . . . . 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 5196	. . . . . . 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 5928	. . . . . 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 4265	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑦𝑅𝑥}
28		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑦 ∈ (dom 𝑅 ∩ 𝐴))
29	28	elin2d 4105	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑦 ∈ 𝐴)
30		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑥 ∈ (dom 𝑅 ∩ 𝐴))
31	30	elin2d 4105	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑥 ∈ 𝐴)
32	31	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → 𝑥 ∈ 𝐴)
33		brinxp 5736	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
34	29, 32, 33	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
35	34	notbid 321	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
36	35	rabbidva 3422	. . . . . . . . . 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 2810	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → (ordTop‘𝑅) ∈ Top)
39	15	adantr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → 𝐴 ∈ 𝑉)
40		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ PosetRel)
41		elinel1 4101	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ dom 𝑅)
42	11	ordtopn1 23545	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
43	40, 41, 42	syl2an 608	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
44		elrestr 17631	. . . . . . . . . 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 1399	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
46	37, 45	eqeltrrd 2865	. . . . . . . 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 7142	. . . . . . 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 6743	. . . . . 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 3947	. . . . 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 3431	. . . . . . . 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 5196	. . . . . . 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 5928	. . . . . 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 4265	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥𝑅𝑦}
54		brinxp 5736	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
55	32, 29, 54	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
56	55	notbid 321	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) ∧ 𝑦 ∈ (dom 𝑅 ∩ 𝐴)) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
57	56	rabbidva 3422	. . . . . . . . . 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 2810	. . . . . . . . 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 23546	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
60	40, 41, 59	syl2an 608	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
61		elrestr 17631	. . . . . . . . . 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 1399	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
63	58, 62	eqeltrrd 2865	. . . . . . . 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 7142	. . . . . . 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 6743	. . . . . 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 3947	. . . . 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 4193	. . . 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 4193	. . 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 23342	. . 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 596	. 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 3947	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  {csn 4588   class class class wbr 5106   ↦ cmpt 5190   × cxp 5654  dom cdm 5656  ran crn 5657  ‘cfv 6561  (class class class)co 7446  ficfi 9432   ↾_t crest 17623  topGenctg 17640  ordTopcordt 17703  PosetRelcps 18770  Topctop 23244  TopOnctopon 23261

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-1o 8507  df-2o 8508  df-en 9005  df-fin 9008  df-fi 9433  df-rest 17625  df-topgen 17646  df-ordt 17705  df-ps 18772  df-top 23245  df-topon 23262  df-bases 23297

This theorem is used by:  ordtrest2  23555