Theorem ordtrest2 21008

Description: An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypotheses

Ref	Expression
ordtrest2.1	⊢ 𝑋 = dom 𝑅
ordtrest2.2	⊢ (𝜑 → 𝑅 ∈ TosetRel )
ordtrest2.3	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
ordtrest2.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴)

Assertion

Ref	Expression
ordtrest2	⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem ordtrest2

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtrest2.2	. . . 4 ⊢ (𝜑 → 𝑅 ∈ TosetRel )
2		tsrps 17221	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ PosetRel)
4		ordtrest2.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
5		dmexg 7097	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
6	1, 5	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
7	4, 6	syl5eqel 2705	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
8		ordtrest2.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
9	7, 8	ssexd 4805	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
10		ordtrest 21006	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ V) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
11	3, 9, 10	syl2anc 693	. 2 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
12		eqid 2622	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})
13		eqid 2622	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})
14	4, 12, 13	ordtval 20993	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
15	1, 14	syl 17	. . . . . 6 ⊢ (𝜑 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
16	15	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
17		fibas 20781	. . . . . 6 ⊢ (fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
18		tgrest 20963	. . . . . 6 ⊢ (((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases ∧ 𝐴 ∈ V) → (topGen‘((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
19	17, 9, 18	sylancr 695	. . . . 5 ⊢ (𝜑 → (topGen‘((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
20	16, 19	eqtr4d 2659	. . . 4 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)))
21		firest 16093	. . . . 5 ⊢ (fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴)) = ((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)
22	21	fveq2i 6194	. . . 4 ⊢ (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴))
23	20, 22	syl6eqr 2674	. . 3 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))))
24		inex1g 4801	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
26		ordttop 21004	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
27	25, 26	syl 17	. . . 4 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
28	4, 12, 13	ordtuni 20994	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
29	1, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
30	29, 7	eqeltrrd 2702	. . . . . . 7 ⊢ (𝜑 → ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
31		uniexb 6973	. . . . . . 7 ⊢ (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ↔ ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
32	30, 31	sylibr 224	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
33		restval 16087	. . . . . 6 ⊢ ((({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ∧ 𝐴 ∈ V) → (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)))
34	32, 9, 33	syl2anc 693	. . . . 5 ⊢ (𝜑 → (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)))
35		sseqin2 3817	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
36	8, 35	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∩ 𝐴) = 𝐴)
37		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
38	37	ordttopon 20997	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
39	25, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
40	4	psssdm 17216	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
41	3, 8, 40	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
42	41	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
43	39, 42	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
44		toponmax 20730	. . . . . . . . . . . 12 ⊢ ((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
45	43, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
46	36, 45	eqeltrd 2701	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
47		elsni 4194	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
48	47	ineq1d 3813	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑋} → (𝑣 ∩ 𝐴) = (𝑋 ∩ 𝐴))
49	48	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑋} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
50	46, 49	syl5ibrcom 237	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝑋} → (𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
51	50	ralrimiv 2965	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ {𝑋} (𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
52		ordtrest2.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴)
53	4, 1, 8, 52	ordtrest2lem 21007	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
54		df-rn 5125	. . . . . . . . . . 11 ⊢ ran 𝑅 = dom ◡𝑅
55		cnvtsr 17222	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
56	1, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ◡𝑅 ∈ TosetRel )
57	4	psrn 17209	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
58	3, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 = ran 𝑅)
59	8, 58	sseqtrd 3641	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ran 𝑅)
60	58	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑋 = ran 𝑅)
61		rabeq 3192	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ran 𝑅 → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)})
62	60, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)})
63		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
64		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
65	63, 64	brcnv 5305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
66		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
67	64, 66	brcnv 5305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧◡𝑅𝑥 ↔ 𝑥𝑅𝑧)
68	65, 67	anbi12ci 734	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥) ↔ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦))
69	68	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ran 𝑅 → ((𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥) ↔ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)))
70	69	rabbiia 3185	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)}
71	62, 70	syl6eqr 2674	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)})
72	71, 52	eqsstr3d 3640	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} ⊆ 𝐴)
73	72	ancom2s 844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} ⊆ 𝐴)
74	54, 56, 59, 73	ordtrest2lem 21007	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴))))
75		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
76	75, 64	brcnv 5305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤◡𝑅𝑧 ↔ 𝑧𝑅𝑤)
77	76	bicomi 214	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝑅𝑤 ↔ 𝑤◡𝑅𝑧)
78	77	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧𝑅𝑤 ↔ 𝑤◡𝑅𝑧))
79	78	notbid 308	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤◡𝑅𝑧))
80	58, 79	rabeqbidv 3195	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})
81	58, 80	mpteq12dv 4733	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧}))
82	81	rneqd 5353	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧}))
83		cnvin 5540	. . . . . . . . . . . . . . 15 ⊢ ◡(𝑅 ∩ (𝐴 × 𝐴)) = (◡𝑅 ∩ ◡(𝐴 × 𝐴))
84		cnvxp 5551	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
85	84	ineq2i 3811	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ∩ ◡(𝐴 × 𝐴)) = (◡𝑅 ∩ (𝐴 × 𝐴))
86	83, 85	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ ◡(𝑅 ∩ (𝐴 × 𝐴)) = (◡𝑅 ∩ (𝐴 × 𝐴))
87	86	fveq2i 6194	. . . . . . . . . . . . 13 ⊢ (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))
88		psss 17214	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
89	3, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
90		ordtcnv 21005	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel → (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
91	89, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
92	87, 91	syl5reqr 2671	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴))))
93	92	eleq2d 2687	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))))
94	82, 93	raleqbidv 3152	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))))
95	74, 94	mpbird 247	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
96		ralunb 3794	. . . . . . . . 9 ⊢ (∀𝑣 ∈ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
97	53, 95, 96	sylanbrc 698	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
98		ralunb 3794	. . . . . . . 8 ⊢ (∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ {𝑋} (𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
99	51, 97, 98	sylanbrc 698	. . . . . . 7 ⊢ (𝜑 → ∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
100		eqid 2622	. . . . . . . 8 ⊢ (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)) = (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴))
101	100	fmpt 6381	. . . . . . 7 ⊢ (∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)):({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
102	99, 101	sylib 208	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)):({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
103		frn 6053	. . . . . 6 ⊢ ((𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)):({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
104	102, 103	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
105	34, 104	eqsstrd 3639	. . . 4 ⊢ (𝜑 → (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
106		tgfiss 20795	. . . 4 ⊢ (((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top ∧ (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
107	27, 105, 106	syl2anc 693	. . 3 ⊢ (𝜑 → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
108	23, 107	eqsstrd 3639	. 2 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
109	11, 108	eqssd 3620	1 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ficfi 8316   ↾_t crest 16081  topGenctg 16098  ordTopcordt 16159  PosetRelcps 17198   TosetRel ctsr 17199  Topctop 20698  TopOnctopon 20715  TopBasesctb 20749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-ordt 16161  df-ps 17200  df-tsr 17201  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  ordtrestixx  21026  cnvordtrestixx  29959