Theorem ordtrest2 23326

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 18639	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ PosetRel)
4		ordtrest2.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
5	1	dmexd 7896	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
6	4, 5	eqeltrid 2873	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
7		ordtrest2.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
8	6, 7	ssexd 5292	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
9		ordtrest 23324	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ V) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
10	3, 8, 9	syl2anc 595	. 2 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
11		eqid 2769	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})
12		eqid 2769	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})
13	4, 11, 12	ordtval 23311	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
14	1, 13	syl 18	. . . . . 6 ⊢ (𝜑 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
15	14	oveq1d 7423	. . . . 5 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
16		fibas 23099	. . . . . 6 ⊢ (fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
17		tgrest 23281	. . . . . 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))
18	16, 8, 17	sylancr 598	. . . . 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))
19	15, 18	eqtr4d 2807	. . . 4 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)))
20		firest 17481	. . . . 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)
21	20	fveq2i 6882	. . . 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))
22	19, 21	eqtr4di 2822	. . 3 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))))
23		inex1g 5287	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
24	1, 23	syl 18	. . . . 5 ⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
25		ordttop 23322	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
26	24, 25	syl 18	. . . 4 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
27	4, 11, 12	ordtuni 23312	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
28	1, 27	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
29	28, 6	eqeltrrd 2870	. . . . . . 7 ⊢ (𝜑 → ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
30		uniexb 7759	. . . . . . 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)
31	29, 30	sylibr 237	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
32		restval 17475	. . . . . 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)))
33	31, 8, 32	syl2anc 595	. . . . 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)))
34		sseqin2 4184	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
35	7, 34	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∩ 𝐴) = 𝐴)
36		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
37	36	ordttopon 23315	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
38	24, 37	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
39	4	psssdm 18634	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
40	3, 7, 39	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
41	40	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
42	38, 41	eleqtrd 2871	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
43		toponmax 23048	. . . . . . . . . . . 12 ⊢ ((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
44	42, 43	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
45	35, 44	eqeltrd 2869	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
46		elsni 4608	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
47	46	ineq1d 4180	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑋} → (𝑣 ∩ 𝐴) = (𝑋 ∩ 𝐴))
48	47	eleq1d 2854	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑋} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
49	45, 48	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝑋} → (𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
50	49	ralrimiv 3162	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ {𝑋} (𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
51		ordtrest2.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴)
52	4, 1, 7, 51	ordtrest2lem 23325	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
53		df-rn 5670	. . . . . . . . . . 11 ⊢ ran 𝑅 = dom ◡𝑅
54		cnvtsr 18640	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
55	1, 54	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ◡𝑅 ∈ TosetRel )
56	4	psrn 18627	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
57	3, 56	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 = ran 𝑅)
58	7, 57	sseqtrd 3981	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ran 𝑅)
59	57	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑋 = ran 𝑅)
60	59	rabeqdv 3438	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)})
61		vex 3467	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
62		vex 3467	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
63	61, 62	brcnv 5866	. . . . . . . . . . . . . . . 16 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
64		vex 3467	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
65	62, 64	brcnv 5866	. . . . . . . . . . . . . . . 16 ⊢ (𝑧◡𝑅𝑥 ↔ 𝑥𝑅𝑧)
66	63, 65	anbi12ci 640	. . . . . . . . . . . . . . 15 ⊢ ((𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥) ↔ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦))
67	66	rabbii 3428	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)}
68	60, 67	eqtr4di 2822	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)})
69	68, 51	eqsstrrd 3980	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} ⊆ 𝐴)
70	69	ancom2s 662	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} ⊆ 𝐴)
71	53, 55, 58, 70	ordtrest2lem 23325	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴))))
72		vex 3467	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
73	72, 62	brcnv 5866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤◡𝑅𝑧 ↔ 𝑧𝑅𝑤)
74	73	bicomi 227	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝑅𝑤 ↔ 𝑤◡𝑅𝑧)
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧𝑅𝑤 ↔ 𝑤◡𝑅𝑧))
76	75	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤◡𝑅𝑧))
77	57, 76	rabeqbidv 3441	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})
78	57, 77	mpteq12dv 5199	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧}))
79	78	rneqd 5926	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧}))
80		psss 18632	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
81	3, 80	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
82		ordtcnv 23323	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel → (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
83	81, 82	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
84		cnvin 6139	. . . . . . . . . . . . . . 15 ⊢ ◡(𝑅 ∩ (𝐴 × 𝐴)) = (◡𝑅 ∩ ◡(𝐴 × 𝐴))
85		cnvxp 6153	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
86	85	ineq2i 4178	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ∩ ◡(𝐴 × 𝐴)) = (◡𝑅 ∩ (𝐴 × 𝐴))
87	84, 86	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ ◡(𝑅 ∩ (𝐴 × 𝐴)) = (◡𝑅 ∩ (𝐴 × 𝐴))
88	87	fveq2i 6882	. . . . . . . . . . . . 13 ⊢ (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))
89	83, 88	eqtr3di 2819	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴))))
90	89	eleq2d 2855	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))))
91	79, 90	raleqbidv 3345	. . . . . . . . . 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)))))
92	71, 91	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
93		ralunb 4158	. . . . . . . . 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)))))
94	52, 92, 93	sylanbrc 594	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
95		ralunb 4158	. . . . . . . 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)))))
96	50, 94, 95	sylanbrc 594	. . . . . . 7 ⊢ (𝜑 → ∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
97		eqid 2769	. . . . . . . 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))
98	97	fmpt 7103	. . . . . . 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))))
99	96, 98	sylib 221	. . . . . 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))))
100	99	frnd 6712	. . . . 5 ⊢ (𝜑 → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣 ∩ 𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
101	33, 100	eqsstrd 3979	. . . 4 ⊢ (𝜑 → (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
102		tgfiss 23113	. . . 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))))
103	26, 101, 102	syl2anc 595	. . 3 ⊢ (𝜑 → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
104	22, 103	eqsstrd 3979	. 2 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
105	10, 104	eqssd 3962	1 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  {csn 4591  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  ficfi 9366   ↾_t crest 17469  topGenctg 17486  ordTopcordt 17549  PosetRelcps 18616   TosetRel ctsr 18617  Topctop 23015  TopOnctopon 23032  TopBasesctb 23067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-1o 8449  df-2o 8450  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9367  df-rest 17471  df-topgen 17492  df-ordt 17551  df-ps 18618  df-tsr 18619  df-top 23016  df-topon 23033  df-bases 23068

This theorem is referenced by:  ordtrestixx  23344  cnvordtrestixx  34244