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Theorem ordtrest2 22438
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 18382 . . . 4 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . . 3 (𝜑𝑅 ∈ PosetRel)
4 ordtrest2.1 . . . . 5 𝑋 = dom 𝑅
51dmexd 7799 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
64, 5eqeltrid 2842 . . . 4 (𝜑𝑋 ∈ V)
7 ordtrest2.3 . . . 4 (𝜑𝐴𝑋)
86, 7ssexd 5263 . . 3 (𝜑𝐴 ∈ V)
9 ordtrest 22436 . . 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 2737 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})
12 eqid 2737 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})
134, 11, 12ordtval 22423 . . . . . . 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 7332 . . . . 5 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
16 fibas 22210 . . . . . 6 (fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
17 tgrest 22393 . . . . . 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 2780 . . . 4 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)))
20 firest 17220 . . . . 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 6815 . . . 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 2795 . . 3 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))))
23 inex1g 5258 . . . . . 6 (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
241, 23syl 17 . . . . 5 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
25 ordttop 22434 . . . . 5 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
2624, 25syl 17 . . . 4 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
274, 11, 12ordtuni 22424 . . . . . . . . 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 7656 . . . . . . 7 (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
3129, 30sylibr 233 . . . . . 6 (𝜑 → ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
32 restval 17214 . . . . . 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 4160 . . . . . . . . . . . 12 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
357, 34sylib 217 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) = 𝐴)
36 eqid 2737 . . . . . . . . . . . . . . 15 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
3736ordttopon 22427 . . . . . . . . . . . . . 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 18377 . . . . . . . . . . . . . . 15 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
403, 7, 39syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
4140fveq2d 6816 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4238, 41eleqtrd 2840 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
43 toponmax 22158 . . . . . . . . . . . 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 4588 . . . . . . . . . . . 12 (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
4746ineq1d 4156 . . . . . . . . . . 11 (𝑣 ∈ {𝑋} → (𝑣𝐴) = (𝑋𝐴))
4847eleq1d 2822 . . . . . . . . . 10 (𝑣 ∈ {𝑋} → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
4945, 48syl5ibrcom 246 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝑋} → (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
5049ralrimiv 3139 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝑋} (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
51 ordtrest2.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)
524, 1, 7, 51ordtrest2lem 22437 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
53 df-rn 5619 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
54 cnvtsr 18383 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
551, 54syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ TosetRel )
564psrn 18370 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
573, 56syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = ran 𝑅)
587, 57sseqtrd 3971 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ran 𝑅)
5957adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑋 = ran 𝑅)
6059rabeqdv 3418 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)})
61 vex 3445 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
62 vex 3445 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
6361, 62brcnv 5812 . . . . . . . . . . . . . . . 16 (𝑦𝑅𝑧𝑧𝑅𝑦)
64 vex 3445 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
6562, 64brcnv 5812 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑥𝑥𝑅𝑧)
6663, 65anbi12ci 628 . . . . . . . . . . . . . . 15 ((𝑦𝑅𝑧𝑧𝑅𝑥) ↔ (𝑥𝑅𝑧𝑧𝑅𝑦))
6766rabbii 3410 . . . . . . . . . . . . . 14 {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)}
6860, 67eqtr4di 2795 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)})
6968, 51eqsstrrd 3970 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7069ancom2s 647 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7153, 55, 58, 70ordtrest2lem 22437 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
72 vex 3445 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
7372, 62brcnv 5812 . . . . . . . . . . . . . . . . 17 (𝑤𝑅𝑧𝑧𝑅𝑤)
7473bicomi 223 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑤𝑤𝑅𝑧)
7574a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑅𝑤𝑤𝑅𝑧))
7675notbid 317 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤𝑅𝑧))
7757, 76rabeqbidv 3420 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})
7857, 77mpteq12dv 5178 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
7978rneqd 5867 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
80 psss 18375 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
813, 80syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
82 ordtcnv 22435 . . . . . . . . . . . . . 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 6071 . . . . . . . . . . . . . . 15 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅(𝐴 × 𝐴))
85 cnvxp 6083 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
8685ineq2i 4154 . . . . . . . . . . . . . . 15 (𝑅(𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8784, 86eqtri 2765 . . . . . . . . . . . . . 14 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8887fveq2i 6815 . . . . . . . . . . . . 13 (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))
8983, 88eqtr3di 2792 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9089eleq2d 2823 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9179, 90raleqbidv 3316 . . . . . . . . . 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 256 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
93 ralunb 4136 . . . . . . . . 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 4136 . . . . . . . 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 2737 . . . . . . . 8 (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) = (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴))
9897fmpt 7024 . . . . . . 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 217 . . . . . 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 6646 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10133, 100eqsstrd 3969 . . . 4 (𝜑 → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
102 tgfiss 22224 . . . 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 3969 . 2 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10510, 104eqssd 3948 1 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3062  {crab 3404  Vcvv 3441  cun 3895  cin 3896  wss 3897  {csn 4571   cuni 4850   class class class wbr 5087  cmpt 5170   × cxp 5606  ccnv 5607  dom cdm 5608  ran crn 5609  wf 6462  cfv 6466  (class class class)co 7317  ficfi 9246  t crest 17208  topGenctg 17225  ordTopcordt 17287  PosetRelcps 18359   TosetRel ctsr 18360  Topctop 22125  TopOnctopon 22142  TopBasesctb 22178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1st 7878  df-2nd 7879  df-1o 8346  df-er 8548  df-en 8784  df-dom 8785  df-fin 8787  df-fi 9247  df-rest 17210  df-topgen 17231  df-ordt 17289  df-ps 18361  df-tsr 18362  df-top 22126  df-topon 22143  df-bases 22179
This theorem is referenced by:  ordtrestixx  22456  cnvordtrestixx  32003
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