Theorem ordtrest 23396

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 5293	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
2	1	adantr 486	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3		eqid 2766	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4		eqid 2766	. . . 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 2766	. . . 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 23383	. . 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 23394	. . . 4 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9		resttop 23354	. . . 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 2766	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
12	11	psssdm2 18662	. . . . . . 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 23387	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
17	16	adantr 486	. . . . . . . 8 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18		toponmax 23120	. . . . . . . 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 17506	. . . . . . 7 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
21	14, 15, 19, 20	syl3anc 1398	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
22	13, 21	eqeltrd 2866	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
23	22	snssd 4757	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
24	13	rabeqdv 3434	. . . . . . . 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 5203	. . . . . . 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 5933	. . . . . 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 4273	. . . . . . . . . 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 4161	. . . . . . . . . . . . 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 4161	. . . . . . . . . . . . . 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 5745	. . . . . . . . . . . . 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 3425	. . . . . . . . . 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 2813	. . . . . . . . 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 4157	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ dom 𝑅)
42	11	ordtopn1 23388	. . . . . . . . . . 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 17506	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
46	37, 45	eqeltrrd 2867	. . . . . . . 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 7117	. . . . . . 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 6721	. . . . . 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 3974	. . . . 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 3434	. . . . . . . 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 5203	. . . . . . 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 5933	. . . . . 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 4273	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥𝑅𝑦}
54		brinxp 5745	. . . . . . . . . . . . 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 3425	. . . . . . . . . 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 2813	. . . . . . . . 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 23389	. . . . . . . . . . 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 17506	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
63	58, 62	eqeltrrd 2867	. . . . . . . 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 7117	. . . . . . 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 6721	. . . . . 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 3974	. . . . 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 4148	. . . 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 4148	. . 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 23185	. . 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 3974	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 2146  {crab 3419  Vcvv 3458   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  {csn 4594   class class class wbr 5114   ↦ cmpt 5197   × cxp 5664  dom cdm 5666  ran crn 5667  ‘cfv 6543  (class class class)co 7423  ficfi 9380   ↾_t crest 17498  topGenctg 17515  ordTopcordt 17578  PosetRelcps 18645  Topctop 23087  TopOnctopon 23104

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-1o 8462  df-2o 8463  df-en 8953  df-fin 8956  df-fi 9381  df-rest 17500  df-topgen 17521  df-ordt 17580  df-ps 18647  df-top 23088  df-topon 23105  df-bases 23140

This theorem is used by:  ordtrest2  23398