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Theorem ordtrest2 23227
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 , but in other sets like 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 (𝜑𝑅 ∈ TosetRel )
ordtrest2.3 (𝜑𝐴𝑋)
ordtrest2.4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 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.
StepHypRef Expression
1 ordtrest2.2 . . . 4 (𝜑𝑅 ∈ TosetRel )
2 tsrps 18644 . . . 4 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . . 3 (𝜑𝑅 ∈ PosetRel)
4 ordtrest2.1 . . . . 5 𝑋 = dom 𝑅
51dmexd 7925 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
64, 5eqeltrid 2842 . . . 4 (𝜑𝑋 ∈ V)
7 ordtrest2.3 . . . 4 (𝜑𝐴𝑋)
86, 7ssexd 5329 . . 3 (𝜑𝐴 ∈ V)
9 ordtrest 23225 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ V) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
103, 8, 9syl2anc 584 . 2 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
11 eqid 2734 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})
12 eqid 2734 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})
134, 11, 12ordtval 23212 . . . . . . 7 (𝑅 ∈ TosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
141, 13syl 17 . . . . . 6 (𝜑 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
1514oveq1d 7445 . . . . 5 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
16 fibas 22999 . . . . . 6 (fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
17 tgrest 23182 . . . . . 6 (((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases ∧ 𝐴 ∈ V) → (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
1816, 8, 17sylancr 587 . . . . 5 (𝜑 → (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
1915, 18eqtr4d 2777 . . . 4 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)))
20 firest 17478 . . . . 5 (fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴)) = ((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)
2120fveq2i 6909 . . . 4 (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴))
2219, 21eqtr4di 2792 . . 3 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))))
23 inex1g 5324 . . . . . 6 (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
241, 23syl 17 . . . . 5 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
25 ordttop 23223 . . . . 5 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
2624, 25syl 17 . . . 4 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
274, 11, 12ordtuni 23213 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
281, 27syl 17 . . . . . . . 8 (𝜑𝑋 = ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
2928, 6eqeltrrd 2839 . . . . . . 7 (𝜑 ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
30 uniexb 7782 . . . . . . 7 (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
3129, 30sylibr 234 . . . . . 6 (𝜑 → ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
32 restval 17472 . . . . . 6 ((({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ∧ 𝐴 ∈ V) → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)))
3331, 8, 32syl2anc 584 . . . . 5 (𝜑 → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)))
34 sseqin2 4230 . . . . . . . . . . . 12 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
357, 34sylib 218 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) = 𝐴)
36 eqid 2734 . . . . . . . . . . . . . . 15 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
3736ordttopon 23216 . . . . . . . . . . . . . 14 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
3824, 37syl 17 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
394psssdm 18639 . . . . . . . . . . . . . . 15 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
403, 7, 39syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
4140fveq2d 6910 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4238, 41eleqtrd 2840 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
43 toponmax 22947 . . . . . . . . . . . 12 ((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
4442, 43syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
4535, 44eqeltrd 2838 . . . . . . . . . 10 (𝜑 → (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
46 elsni 4647 . . . . . . . . . . . 12 (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
4746ineq1d 4226 . . . . . . . . . . 11 (𝑣 ∈ {𝑋} → (𝑣𝐴) = (𝑋𝐴))
4847eleq1d 2823 . . . . . . . . . 10 (𝑣 ∈ {𝑋} → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
4945, 48syl5ibrcom 247 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝑋} → (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
5049ralrimiv 3142 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝑋} (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
51 ordtrest2.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)
524, 1, 7, 51ordtrest2lem 23226 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
53 df-rn 5699 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
54 cnvtsr 18645 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
551, 54syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ TosetRel )
564psrn 18632 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
573, 56syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = ran 𝑅)
587, 57sseqtrd 4035 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ran 𝑅)
5957adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑋 = ran 𝑅)
6059rabeqdv 3448 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)})
61 vex 3481 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
62 vex 3481 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
6361, 62brcnv 5895 . . . . . . . . . . . . . . . 16 (𝑦𝑅𝑧𝑧𝑅𝑦)
64 vex 3481 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
6562, 64brcnv 5895 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑥𝑥𝑅𝑧)
6663, 65anbi12ci 629 . . . . . . . . . . . . . . 15 ((𝑦𝑅𝑧𝑧𝑅𝑥) ↔ (𝑥𝑅𝑧𝑧𝑅𝑦))
6766rabbii 3438 . . . . . . . . . . . . . 14 {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)}
6860, 67eqtr4di 2792 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)})
6968, 51eqsstrrd 4034 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7069ancom2s 650 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7153, 55, 58, 70ordtrest2lem 23226 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
72 vex 3481 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
7372, 62brcnv 5895 . . . . . . . . . . . . . . . . 17 (𝑤𝑅𝑧𝑧𝑅𝑤)
7473bicomi 224 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑤𝑤𝑅𝑧)
7574a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑅𝑤𝑤𝑅𝑧))
7675notbid 318 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤𝑅𝑧))
7757, 76rabeqbidv 3451 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})
7857, 77mpteq12dv 5238 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
7978rneqd 5951 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
80 psss 18637 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
813, 80syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
82 ordtcnv 23224 . . . . . . . . . . . . . 14 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
8381, 82syl 17 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
84 cnvin 6166 . . . . . . . . . . . . . . 15 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅(𝐴 × 𝐴))
85 cnvxp 6178 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
8685ineq2i 4224 . . . . . . . . . . . . . . 15 (𝑅(𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8784, 86eqtri 2762 . . . . . . . . . . . . . 14 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8887fveq2i 6909 . . . . . . . . . . . . 13 (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))
8983, 88eqtr3di 2789 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9089eleq2d 2824 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9179, 90raleqbidv 3343 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9271, 91mpbird 257 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
93 ralunb 4206 . . . . . . . . 9 (∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9452, 92, 93sylanbrc 583 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
95 ralunb 4206 . . . . . . . 8 (∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ {𝑋} (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9650, 94, 95sylanbrc 583 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
97 eqid 2734 . . . . . . . 8 (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) = (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴))
9897fmpt 7129 . . . . . . 7 (∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)):({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9996, 98sylib 218 . . . . . 6 (𝜑 → (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)):({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10099frnd 6744 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10133, 100eqsstrd 4033 . . . 4 (𝜑 → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
102 tgfiss 23013 . . . 4 (((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top ∧ (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10326, 101, 102syl2anc 584 . . 3 (𝜑 → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10422, 103eqsstrd 4033 . 2 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10510, 104eqssd 4012 1 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  cun 3960  cin 3961  wss 3962  {csn 4630   cuni 4911   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  wf 6558  cfv 6562  (class class class)co 7430  ficfi 9447  t crest 17466  topGenctg 17483  ordTopcordt 17545  PosetRelcps 18621   TosetRel ctsr 18622  Topctop 22914  TopOnctopon 22931  TopBasesctb 22967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-1o 8504  df-2o 8505  df-en 8984  df-dom 8985  df-fin 8987  df-fi 9448  df-rest 17468  df-topgen 17489  df-ordt 17547  df-ps 18623  df-tsr 18624  df-top 22915  df-topon 22932  df-bases 22968
This theorem is referenced by:  ordtrestixx  23245  cnvordtrestixx  33873
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