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Theorem ordtrest2NEW 29775
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.) (Revised by Thierry Arnoux, 11-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b 𝐵 = (Base‘𝐾)
ordtNEW.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
ordtrest2NEW.2 (𝜑𝐾 ∈ Toset)
ordtrest2NEW.3 (𝜑𝐴𝐵)
ordtrest2NEW.4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
Assertion
Ref Expression
ordtrest2NEW (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
Distinct variable groups:   𝑥,𝑦,   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦   𝑥,𝐴,𝑦,𝑧   𝑧,   𝑧,𝐴   𝑧,𝐵   𝜑,𝑥,𝑦,𝑧   𝑧,𝐾

Proof of Theorem ordtrest2NEW
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtrest2NEW.2 . . . 4 (𝜑𝐾 ∈ Toset)
2 tospos 29467 . . . 4 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3 posprs 16881 . . . 4 (𝐾 ∈ Poset → 𝐾 ∈ Preset )
41, 2, 33syl 18 . . 3 (𝜑𝐾 ∈ Preset )
5 ordtrest2NEW.3 . . 3 (𝜑𝐴𝐵)
6 ordtNEW.b . . . 4 𝐵 = (Base‘𝐾)
7 ordtNEW.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
86, 7ordtrestNEW 29773 . . 3 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
94, 5, 8syl2anc 692 . 2 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
10 eqid 2621 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
11 eqid 2621 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})
126, 7, 10, 11ordtprsval 29770 . . . . . . 7 (𝐾 ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))))
134, 12syl 17 . . . . . 6 (𝜑 → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))))
1413oveq1d 6625 . . . . 5 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
15 fibas 20705 . . . . . 6 (fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases
16 fvex 6163 . . . . . . . . 9 (Base‘𝐾) ∈ V
176, 16eqeltri 2694 . . . . . . . 8 𝐵 ∈ V
1817a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
1918, 5ssexd 4770 . . . . . 6 (𝜑𝐴 ∈ V)
20 tgrest 20886 . . . . . 6 (((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases ∧ 𝐴 ∈ V) → (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
2115, 19, 20sylancr 694 . . . . 5 (𝜑 → (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
2214, 21eqtr4d 2658 . . . 4 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)))
23 firest 16025 . . . . 5 (fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴)) = ((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)
2423fveq2i 6156 . . . 4 (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴))
2522, 24syl6eqr 2673 . . 3 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))))
26 fvex 6163 . . . . . . . 8 (le‘𝐾) ∈ V
2726inex1 4764 . . . . . . 7 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
287, 27eqeltri 2694 . . . . . 6 ∈ V
2928inex1 4764 . . . . 5 ( ∩ (𝐴 × 𝐴)) ∈ V
30 ordttop 20927 . . . . 5 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
3129, 30mp1i 13 . . . 4 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
326, 7, 10, 11ordtprsuni 29771 . . . . . . . . 9 (𝐾 ∈ Preset → 𝐵 = ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))
334, 32syl 17 . . . . . . . 8 (𝜑𝐵 = ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))
3433, 18eqeltrrd 2699 . . . . . . 7 (𝜑 ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
35 uniexb 6928 . . . . . . 7 (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V ↔ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
3634, 35sylibr 224 . . . . . 6 (𝜑 → ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
37 restval 16019 . . . . . 6 ((({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V ∧ 𝐴 ∈ V) → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)))
3836, 19, 37syl2anc 692 . . . . 5 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)))
39 sseqin2 3800 . . . . . . . . . . . 12 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
405, 39sylib 208 . . . . . . . . . . 11 (𝜑 → (𝐵𝐴) = 𝐴)
41 eqid 2621 . . . . . . . . . . . . . . 15 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
4241ordttopon 20920 . . . . . . . . . . . . . 14 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ∩ (𝐴 × 𝐴))))
4329, 42mp1i 13 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ∩ (𝐴 × 𝐴))))
446, 7prsssdm 29769 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
454, 5, 44syl2anc 692 . . . . . . . . . . . . . 14 (𝜑 → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
4645fveq2d 6157 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom ( ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4743, 46eleqtrd 2700 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
48 toponmax 20652 . . . . . . . . . . . 12 ((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4947, 48syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
5040, 49eqeltrd 2698 . . . . . . . . . 10 (𝜑 → (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
51 elsni 4170 . . . . . . . . . . . 12 (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
5251ineq1d 3796 . . . . . . . . . . 11 (𝑣 ∈ {𝐵} → (𝑣𝐴) = (𝐵𝐴))
5352eleq1d 2683 . . . . . . . . . 10 (𝑣 ∈ {𝐵} → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5450, 53syl5ibrcom 237 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐵} → (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5554ralrimiv 2960 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
56 ordtrest2NEW.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
576, 7, 1, 5, 56ordtrest2NEWlem 29774 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
58 eqid 2621 . . . . . . . . . . . 12 (ODual‘𝐾) = (ODual‘𝐾)
5958, 6odubas 17065 . . . . . . . . . . 11 𝐵 = (Base‘(ODual‘𝐾))
607cnveqi 5262 . . . . . . . . . . . 12 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
61 cnvin 5504 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
62 cnvxp 5515 . . . . . . . . . . . . . 14 (𝐵 × 𝐵) = (𝐵 × 𝐵)
6362ineq2i 3794 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
64 eqid 2621 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
6558, 64oduleval 17063 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘(ODual‘𝐾))
6665ineq1i 3793 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6761, 63, 663eqtri 2647 . . . . . . . . . . . 12 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6860, 67eqtri 2643 . . . . . . . . . . 11 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6958odutos 29472 . . . . . . . . . . . 12 (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
701, 69syl 17 . . . . . . . . . . 11 (𝜑 → (ODual‘𝐾) ∈ Toset)
71 vex 3192 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
72 vex 3192 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
7371, 72brcnv 5270 . . . . . . . . . . . . . . . 16 (𝑦 𝑧𝑧 𝑦)
74 vex 3192 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
7572, 74brcnv 5270 . . . . . . . . . . . . . . . 16 (𝑧 𝑥𝑥 𝑧)
7673, 75anbi12ci 733 . . . . . . . . . . . . . . 15 ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦))
7776a1i 11 . . . . . . . . . . . . . 14 (𝑧𝐵 → ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦)))
7877rabbiia 3176 . . . . . . . . . . . . 13 {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} = {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)}
7978, 56syl5eqss 3633 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
8079ancom2s 843 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
8159, 68, 70, 5, 80ordtrest2NEWlem 29774 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
82 vex 3192 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
8382, 72brcnv 5270 . . . . . . . . . . . . . . . . 17 (𝑤 𝑧𝑧 𝑤)
8483bicomi 214 . . . . . . . . . . . . . . . 16 (𝑧 𝑤𝑤 𝑧)
8584a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 𝑤𝑤 𝑧))
8685notbid 308 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧 𝑤 ↔ ¬ 𝑤 𝑧))
8786rabbidv 3180 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝐵 ∣ ¬ 𝑧 𝑤} = {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
8887mpteq2dv 4710 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
8988rneqd 5318 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
90 cnvin 5504 . . . . . . . . . . . . . . 15 ( ∩ (𝐴 × 𝐴)) = ( (𝐴 × 𝐴))
91 cnvxp 5515 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
9291ineq2i 3794 . . . . . . . . . . . . . . 15 ( (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9390, 92eqtri 2643 . . . . . . . . . . . . . 14 ( ∩ (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9493fveq2i 6156 . . . . . . . . . . . . 13 (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴)))
956ressprs 29464 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )
964, 5, 95syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾s 𝐴) ∈ Preset )
97 eqid 2621 . . . . . . . . . . . . . . . 16 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
98 eqid 2621 . . . . . . . . . . . . . . . 16 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
9997, 98ordtcnvNEW 29772 . . . . . . . . . . . . . . 15 ((𝐾s 𝐴) ∈ Preset → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
10096, 99syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
1016, 7prsss 29768 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
1024, 5, 101syl2anc 692 . . . . . . . . . . . . . . . . 17 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
103 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝐾s 𝐴) = (𝐾s 𝐴)
104103, 64ressle 15991 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
10519, 104syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
106103, 6ressbas2 15863 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
1075, 106syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = (Base‘(𝐾s 𝐴)))
108107sqxpeqd 5106 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
109105, 108ineq12d 3798 . . . . . . . . . . . . . . . . 17 (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
110102, 109eqtrd 2655 . . . . . . . . . . . . . . . 16 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
111110cnveqd 5263 . . . . . . . . . . . . . . 15 (𝜑( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
112111fveq2d 6157 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
113110fveq2d 6157 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
114100, 112, 1133eqtr4d 2665 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
11594, 114syl5reqr 2670 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
116115eleq2d 2684 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11789, 116raleqbidv 3144 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11881, 117mpbird 247 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
119 ralunb 3777 . . . . . . . . 9 (∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
12057, 118, 119sylanbrc 697 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
121 ralunb 3777 . . . . . . . 8 (∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
12255, 120, 121sylanbrc 697 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
123 eqid 2621 . . . . . . . 8 (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) = (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴))
124123fmpt 6342 . . . . . . 7 (∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)):({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))⟶(ordTop‘( ∩ (𝐴 × 𝐴))))
125122, 124sylib 208 . . . . . 6 (𝜑 → (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)):({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))⟶(ordTop‘( ∩ (𝐴 × 𝐴))))
126 frn 6015 . . . . . 6 ((𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)):({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))⟶(ordTop‘( ∩ (𝐴 × 𝐴))) → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
127125, 126syl 17 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12838, 127eqsstrd 3623 . . . 4 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
129 tgfiss 20719 . . . 4 (((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top ∧ (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴)))) → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
13031, 128, 129syl2anc 692 . . 3 (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
13125, 130eqsstrd 3623 . 2 (𝜑 → ((ordTop‘ ) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
1329, 131eqssd 3604 1 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cun 3557  cin 3558  wss 3559  {csn 4153   cuni 4407   class class class wbr 4618  cmpt 4678   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  wf 5848  cfv 5852  (class class class)co 6610  ficfi 8268  Basecbs 15792  s cress 15793  lecple 15880  t crest 16013  topGenctg 16030  ordTopcordt 16091   Preset cpreset 16858  Posetcpo 16872  Tosetctos 16965  ODualcodu 17060  Topctop 20630  TopOnctopon 20647  TopBasesctb 20673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fi 8269  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-dec 11446  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-ple 15893  df-rest 16015  df-topgen 16036  df-ordt 16093  df-preset 16860  df-poset 16878  df-toset 16966  df-odu 17061  df-top 20631  df-topon 20648  df-bases 20674
This theorem is referenced by: (None)
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