Theorem ordtrest2 23091

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 18546	. . . 4 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ PosetRel)
4		ordtrest2.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
5	1	dmexd 7879	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
6	4, 5	eqeltrid 2832	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
7		ordtrest2.3	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
8	6, 7	ssexd 5279	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
9		ordtrest 23089	. . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ V) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
10	3, 8, 9	syl2anc 584	. 2 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
11		eqid 2729	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})
12		eqid 2729	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})
13	4, 11, 12	ordtval 23076	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
14	1, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
15	14	oveq1d 7402	. . . . 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 22864	. . . . . 6 ⊢ (fi‘({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
17		tgrest 23046	. . . . . 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 587	. . . . 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 2767	. . . 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 17395	. . . . 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 6861	. . . 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 2782	. . 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 5274	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
24	1, 23	syl 17	. . . . 5 ⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
25		ordttop 23087	. . . . 5 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
26	24, 25	syl 17	. . . 4 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
27	4, 11, 12	ordtuni 23077	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
28	1, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
29	28, 6	eqeltrrd 2829	. . . . . . 7 ⊢ (𝜑 → ∪ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
30		uniexb 7740	. . . . . . 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 234	. . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
32		restval 17389	. . . . . 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 584	. . . . 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 4186	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
35	7, 34	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∩ 𝐴) = 𝐴)
36		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
37	36	ordttopon 23080	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
38	24, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
39	4	psssdm 18541	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
40	3, 7, 39	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
41	40	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
42	38, 41	eleqtrd 2830	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
43		toponmax 22813	. . . . . . . . . . . 12 ⊢ ((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
45	35, 44	eqeltrd 2828	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
46		elsni 4606	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
47	46	ineq1d 4182	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝑋} → (𝑣 ∩ 𝐴) = (𝑋 ∩ 𝐴))
48	47	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝑋} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
49	45, 48	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝑋} → (𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
50	49	ralrimiv 3124	. . . . . . . 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 23090	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
53		df-rn 5649	. . . . . . . . . . 11 ⊢ ran 𝑅 = dom ◡𝑅
54		cnvtsr 18547	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
55	1, 54	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ◡𝑅 ∈ TosetRel )
56	4	psrn 18534	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
57	3, 56	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 = ran 𝑅)
58	7, 57	sseqtrd 3983	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ran 𝑅)
59	57	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑋 = ran 𝑅)
60	59	rabeqdv 3421	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)})
61		vex 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
62		vex 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
63	61, 62	brcnv 5846	. . . . . . . . . . . . . . . 16 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
64		vex 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
65	62, 64	brcnv 5846	. . . . . . . . . . . . . . . 16 ⊢ (𝑧◡𝑅𝑥 ↔ 𝑥𝑅𝑧)
66	63, 65	anbi12ci 629	. . . . . . . . . . . . . . 15 ⊢ ((𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥) ↔ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦))
67	66	rabbii 3411	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)}
68	60, 67	eqtr4di 2782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)})
69	68, 51	eqsstrrd 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} ⊆ 𝐴)
70	69	ancom2s 650	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦◡𝑅𝑧 ∧ 𝑧◡𝑅𝑥)} ⊆ 𝐴)
71	53, 55, 58, 70	ordtrest2lem 23090	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴))))
72		vex 3451	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
73	72, 62	brcnv 5846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤◡𝑅𝑧 ↔ 𝑧𝑅𝑤)
74	73	bicomi 224	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝑅𝑤 ↔ 𝑤◡𝑅𝑧)
75	74	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧𝑅𝑤 ↔ 𝑤◡𝑅𝑧))
76	75	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤◡𝑅𝑧))
77	57, 76	rabeqbidv 3424	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧})
78	57, 77	mpteq12dv 5194	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧}))
79	78	rneqd 5902	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤◡𝑅𝑧}))
80		psss 18539	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
81	3, 80	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
82		ordtcnv 23088	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel → (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
83	81, 82	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
84		cnvin 6117	. . . . . . . . . . . . . . 15 ⊢ ◡(𝑅 ∩ (𝐴 × 𝐴)) = (◡𝑅 ∩ ◡(𝐴 × 𝐴))
85		cnvxp 6130	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
86	85	ineq2i 4180	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ∩ ◡(𝐴 × 𝐴)) = (◡𝑅 ∩ (𝐴 × 𝐴))
87	84, 86	eqtri 2752	. . . . . . . . . . . . . 14 ⊢ ◡(𝑅 ∩ (𝐴 × 𝐴)) = (◡𝑅 ∩ (𝐴 × 𝐴))
88	87	fveq2i 6861	. . . . . . . . . . . . 13 ⊢ (ordTop‘◡(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))
89	83, 88	eqtr3di 2779	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴))))
90	89	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∩ 𝐴) ∈ (ordTop‘(◡𝑅 ∩ (𝐴 × 𝐴)))))
91	79, 90	raleqbidv 3319	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
93		ralunb 4160	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
95		ralunb 4160	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → ∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
97		eqid 2729	. . . . . . . 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 7082	. . . . . . 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 218	. . . . . 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 6696	. . . . 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 3981	. . . 4 ⊢ (𝜑 → (({𝑋} ∪ (ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
102		tgfiss 22878	. . . 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 584	. . 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 3981	. 2 ⊢ (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
105	10, 104	eqssd 3964	1 ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ficfi 9361   ↾_t crest 17383  topGenctg 17400  ordTopcordt 17462  PosetRelcps 18523   TosetRel ctsr 18524  Topctop 22780  TopOnctopon 22797  TopBasesctb 22832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-2o 8435  df-en 8919  df-dom 8920  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-ordt 17464  df-ps 18525  df-tsr 18526  df-top 22781  df-topon 22798  df-bases 22833

This theorem is referenced by:  ordtrestixx  23109  cnvordtrestixx  33903