Theorem ordtrest 23177

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 5256	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
3		eqid 2737	. . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
4		eqid 2737	. . . 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 2737	. . . 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 23164	. . 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 23175	. . . 4 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Top)
9		resttop 23135	. . . 4 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
10	8, 9	sylan 581	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → ((ordTop‘𝑅) ↾t 𝐴) ∈ Top)
11		eqid 2737	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
12	11	psssdm2 18538	. . . . . . 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 23168	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅))
18		toponmax 22901	. . . . . . . 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 17382	. . . . . . 7 ⊢ (((ordTop‘𝑅) ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ dom 𝑅 ∈ (ordTop‘𝑅)) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
21	14, 15, 19, 20	syl3anc 1374	. . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (dom 𝑅 ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
22	13, 21	eqeltrd 2837	. . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → dom (𝑅 ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘𝑅) ↾t 𝐴))
23	22	snssd 4753	. . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → {dom (𝑅 ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘𝑅) ↾t 𝐴))
24	13	rabeqdv 3405	. . . . . . . 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 5173	. . . . . . 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 5887	. . . . . 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 4258	. . . . . . . . . 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 4146	. . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . 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 5703	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
34	29, 32, 33	syl2anc 585	. . . . . . . . . . . 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 3396	. . . . . . . . . 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 2784	. . . . . . . . 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 4142	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ dom 𝑅)
42	11	ordtopn1 23169	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
43	40, 41, 42	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∈ (ordTop‘𝑅))
44		elrestr 17382	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
46	37, 45	eqeltrrd 2838	. . . . . . . 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 7061	. . . . . . 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 6670	. . . . . 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 3957	. . . . 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 3405	. . . . . . . 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 5173	. . . . . . 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 5887	. . . . . 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 4258	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝑅 ∩ 𝐴) ∣ ¬ 𝑥𝑅𝑦}
54		brinxp 5703	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
55	32, 29, 54	syl2anc 585	. . . . . . . . . . . 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 3396	. . . . . . . . . 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 2784	. . . . . . . . 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 23170	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
60	40, 41, 59	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∈ (ordTop‘𝑅))
61		elrestr 17382	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ (((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (dom 𝑅 ∩ 𝐴)) → ({𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦} ∩ 𝐴) ∈ ((ordTop‘𝑅) ↾t 𝐴))
63	58, 62	eqeltrrd 2838	. . . . . . . 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 7061	. . . . . . 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 6670	. . . . . 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 3957	. . . . 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 4133	. . . 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 4133	. . 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 22966	. . 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 585	. 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 3957	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   × cxp 5622  dom cdm 5624  ran crn 5625  ‘cfv 6492  (class class class)co 7360  ficfi 9316   ↾_t crest 17374  topGenctg 17391  ordTopcordt 17454  PosetRelcps 18521  Topctop 22868  TopOnctopon 22885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8398  df-2o 8399  df-en 8887  df-fin 8890  df-fi 9317  df-rest 17376  df-topgen 17397  df-ordt 17456  df-ps 18523  df-top 22869  df-topon 22886  df-bases 22921

This theorem is referenced by:  ordtrest2  23179