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Theorem ordtrest2 21816
 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 (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 17825 . . . 4 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . . 3 (𝜑𝑅 ∈ PosetRel)
4 ordtrest2.1 . . . . 5 𝑋 = dom 𝑅
51dmexd 7598 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
64, 5eqeltrid 2894 . . . 4 (𝜑𝑋 ∈ V)
7 ordtrest2.3 . . . 4 (𝜑𝐴𝑋)
86, 7ssexd 5192 . . 3 (𝜑𝐴 ∈ V)
9 ordtrest 21814 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ V) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
103, 8, 9syl2anc 587 . 2 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
11 eqid 2798 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})
12 eqid 2798 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})
134, 11, 12ordtval 21801 . . . . . . 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 7150 . . . . 5 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
16 fibas 21589 . . . . . 6 (fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
17 tgrest 21771 . . . . . 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 590 . . . . 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 2836 . . . 4 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)))
20 firest 16700 . . . . 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 6648 . . . 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 2851 . . 3 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))))
23 inex1g 5187 . . . . . 6 (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
241, 23syl 17 . . . . 5 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
25 ordttop 21812 . . . . 5 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
2624, 25syl 17 . . . 4 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
274, 11, 12ordtuni 21802 . . . . . . . . 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 2891 . . . . . . 7 (𝜑 ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
30 uniexb 7468 . . . . . . 7 (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
3129, 30sylibr 237 . . . . . 6 (𝜑 → ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
32 restval 16694 . . . . . 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 587 . . . . 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 4142 . . . . . . . . . . . 12 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
357, 34sylib 221 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) = 𝐴)
36 eqid 2798 . . . . . . . . . . . . . . 15 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
3736ordttopon 21805 . . . . . . . . . . . . . 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 17820 . . . . . . . . . . . . . . 15 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
403, 7, 39syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
4140fveq2d 6649 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4238, 41eleqtrd 2892 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
43 toponmax 21538 . . . . . . . . . . . 12 ((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
4442, 43syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
4535, 44eqeltrd 2890 . . . . . . . . . 10 (𝜑 → (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
46 elsni 4542 . . . . . . . . . . . 12 (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
4746ineq1d 4138 . . . . . . . . . . 11 (𝑣 ∈ {𝑋} → (𝑣𝐴) = (𝑋𝐴))
4847eleq1d 2874 . . . . . . . . . 10 (𝑣 ∈ {𝑋} → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
4945, 48syl5ibrcom 250 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝑋} → (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
5049ralrimiv 3148 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝑋} (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
51 ordtrest2.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)
524, 1, 7, 51ordtrest2lem 21815 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
53 df-rn 5530 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
54 cnvtsr 17826 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
551, 54syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ TosetRel )
564psrn 17813 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
573, 56syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = ran 𝑅)
587, 57sseqtrd 3955 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ran 𝑅)
5957adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑋 = ran 𝑅)
6059rabeqdv 3432 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)})
61 vex 3444 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
62 vex 3444 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
6361, 62brcnv 5717 . . . . . . . . . . . . . . . 16 (𝑦𝑅𝑧𝑧𝑅𝑦)
64 vex 3444 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
6562, 64brcnv 5717 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑥𝑥𝑅𝑧)
6663, 65anbi12ci 630 . . . . . . . . . . . . . . 15 ((𝑦𝑅𝑧𝑧𝑅𝑥) ↔ (𝑥𝑅𝑧𝑧𝑅𝑦))
6766rabbii 3420 . . . . . . . . . . . . . 14 {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)}
6860, 67eqtr4di 2851 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)})
6968, 51eqsstrrd 3954 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7069ancom2s 649 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7153, 55, 58, 70ordtrest2lem 21815 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
72 vex 3444 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
7372, 62brcnv 5717 . . . . . . . . . . . . . . . . 17 (𝑤𝑅𝑧𝑧𝑅𝑤)
7473bicomi 227 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑤𝑤𝑅𝑧)
7574a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑅𝑤𝑤𝑅𝑧))
7675notbid 321 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤𝑅𝑧))
7757, 76rabeqbidv 3433 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})
7857, 77mpteq12dv 5115 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
7978rneqd 5772 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
80 cnvin 5970 . . . . . . . . . . . . . . 15 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅(𝐴 × 𝐴))
81 cnvxp 5981 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
8281ineq2i 4136 . . . . . . . . . . . . . . 15 (𝑅(𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8380, 82eqtri 2821 . . . . . . . . . . . . . 14 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8483fveq2i 6648 . . . . . . . . . . . . 13 (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))
85 psss 17818 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
863, 85syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
87 ordtcnv 21813 . . . . . . . . . . . . . 14 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
8886, 87syl 17 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
8984, 88syl5reqr 2848 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9089eleq2d 2875 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9179, 90raleqbidv 3354 . . . . . . . . . 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 260 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
93 ralunb 4118 . . . . . . . . 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 586 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
95 ralunb 4118 . . . . . . . 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 586 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
97 eqid 2798 . . . . . . . 8 (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) = (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴))
9897fmpt 6851 . . . . . . 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 221 . . . . . 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 6494 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10133, 100eqsstrd 3953 . . . 4 (𝜑 → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
102 tgfiss 21603 . . . 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 587 . . 3 (𝜑 → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10422, 103eqsstrd 3953 . 2 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10510, 104eqssd 3932 1 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  {csn 4525  ∪ cuni 4800   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517  ◡ccnv 5518  dom cdm 5519  ran crn 5520  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ficfi 8860   ↾t crest 16688  topGenctg 16705  ordTopcordt 16766  PosetRelcps 17802   TosetRel ctsr 17803  Topctop 21505  TopOnctopon 21522  TopBasesctb 21557 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-fin 8498  df-fi 8861  df-rest 16690  df-topgen 16711  df-ordt 16768  df-ps 17804  df-tsr 17805  df-top 21506  df-topon 21523  df-bases 21558 This theorem is referenced by:  ordtrestixx  21834  cnvordtrestixx  31278
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