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Theorem ordtrest2NEW 30326
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 30015 . . . 4 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3 posprs 17177 . . . 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 30324 . . 3 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
94, 5, 8syl2anc 574 . 2 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
10 eqid 2774 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
11 eqid 2774 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})
126, 7, 10, 11ordtprsval 30321 . . . . . . 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 6827 . . . . 5 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
15 fibas 21022 . . . . . 6 (fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases
166fvexi 6360 . . . . . . . 8 𝐵 ∈ V
1716a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
1817, 5ssexd 4953 . . . . . 6 (𝜑𝐴 ∈ V)
19 tgrest 21204 . . . . . 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))
2015, 18, 19sylancr 576 . . . . 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))
2114, 20eqtr4d 2811 . . . 4 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)))
22 firest 16321 . . . . 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)
2322fveq2i 6351 . . . 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))
2421, 23syl6eqr 2826 . . 3 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))))
25 fvex 6359 . . . . . . . 8 (le‘𝐾) ∈ V
2625inex1 4947 . . . . . . 7 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
277, 26eqeltri 2849 . . . . . 6 ∈ V
2827inex1 4947 . . . . 5 ( ∩ (𝐴 × 𝐴)) ∈ V
29 ordttop 21245 . . . . 5 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
3028, 29mp1i 13 . . . 4 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
316, 7, 10, 11ordtprsuni 30322 . . . . . . . . 9 (𝐾 ∈ Preset → 𝐵 = ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))
324, 31syl 17 . . . . . . . 8 (𝜑𝐵 = ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))
3332, 17eqeltrrd 2854 . . . . . . 7 (𝜑 ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
34 uniexb 7141 . . . . . . 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)
3533, 34sylibr 225 . . . . . 6 (𝜑 → ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
36 restval 16315 . . . . . 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)))
3735, 18, 36syl2anc 574 . . . . 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)))
38 sseqin2 3975 . . . . . . . . . . . 12 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
395, 38sylib 209 . . . . . . . . . . 11 (𝜑 → (𝐵𝐴) = 𝐴)
40 eqid 2774 . . . . . . . . . . . . . . 15 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
4140ordttopon 21238 . . . . . . . . . . . . . 14 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ∩ (𝐴 × 𝐴))))
4228, 41mp1i 13 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ∩ (𝐴 × 𝐴))))
436, 7prsssdm 30320 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
444, 5, 43syl2anc 574 . . . . . . . . . . . . . 14 (𝜑 → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
4544fveq2d 6352 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom ( ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4642, 45eleqtrd 2855 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
47 toponmax 20971 . . . . . . . . . . . 12 ((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4846, 47syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4939, 48eqeltrd 2853 . . . . . . . . . 10 (𝜑 → (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
50 elsni 4343 . . . . . . . . . . . 12 (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
5150ineq1d 3971 . . . . . . . . . . 11 (𝑣 ∈ {𝐵} → (𝑣𝐴) = (𝐵𝐴))
5251eleq1d 2838 . . . . . . . . . 10 (𝑣 ∈ {𝐵} → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5349, 52syl5ibrcom 238 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐵} → (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5453ralrimiv 3117 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
55 ordtrest2NEW.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
566, 7, 1, 5, 55ordtrest2NEWlem 30325 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
57 eqid 2774 . . . . . . . . . . . 12 (ODual‘𝐾) = (ODual‘𝐾)
5857, 6odubas 17361 . . . . . . . . . . 11 𝐵 = (Base‘(ODual‘𝐾))
597cnveqi 5447 . . . . . . . . . . . 12 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
60 cnvin 5692 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
61 cnvxp 5703 . . . . . . . . . . . . . 14 (𝐵 × 𝐵) = (𝐵 × 𝐵)
6261ineq2i 3969 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
63 eqid 2774 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
6457, 63oduleval 17359 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘(ODual‘𝐾))
6564ineq1i 3968 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6660, 62, 653eqtri 2800 . . . . . . . . . . . 12 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6759, 66eqtri 2796 . . . . . . . . . . 11 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6857odutos 30020 . . . . . . . . . . . 12 (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
691, 68syl 17 . . . . . . . . . . 11 (𝜑 → (ODual‘𝐾) ∈ Toset)
70 vex 3358 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
71 vex 3358 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
7270, 71brcnv 5455 . . . . . . . . . . . . . . 15 (𝑦 𝑧𝑧 𝑦)
73 vex 3358 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
7471, 73brcnv 5455 . . . . . . . . . . . . . . 15 (𝑧 𝑥𝑥 𝑧)
7572, 74anbi12ci 614 . . . . . . . . . . . . . 14 ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦))
7675rabbii 3339 . . . . . . . . . . . . 13 {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} = {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)}
7776, 55syl5eqss 3805 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
7877ancom2s 630 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
7958, 67, 69, 5, 78ordtrest2NEWlem 30325 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
80 vex 3358 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
8180, 71brcnv 5455 . . . . . . . . . . . . . . . . 17 (𝑤 𝑧𝑧 𝑤)
8281bicomi 215 . . . . . . . . . . . . . . . 16 (𝑧 𝑤𝑤 𝑧)
8382a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 𝑤𝑤 𝑧))
8483notbid 308 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧 𝑤 ↔ ¬ 𝑤 𝑧))
8584rabbidv 3343 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝐵 ∣ ¬ 𝑧 𝑤} = {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
8685mpteq2dv 4892 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
8786rneqd 5503 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
88 cnvin 5692 . . . . . . . . . . . . . . 15 ( ∩ (𝐴 × 𝐴)) = ( (𝐴 × 𝐴))
89 cnvxp 5703 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
9089ineq2i 3969 . . . . . . . . . . . . . . 15 ( (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9188, 90eqtri 2796 . . . . . . . . . . . . . 14 ( ∩ (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9291fveq2i 6351 . . . . . . . . . . . . 13 (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴)))
936ressprs 30012 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )
944, 5, 93syl2anc 574 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾s 𝐴) ∈ Preset )
95 eqid 2774 . . . . . . . . . . . . . . . 16 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
96 eqid 2774 . . . . . . . . . . . . . . . 16 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
9795, 96ordtcnvNEW 30323 . . . . . . . . . . . . . . 15 ((𝐾s 𝐴) ∈ Preset → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
9894, 97syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
996, 7prsss 30319 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
1004, 5, 99syl2anc 574 . . . . . . . . . . . . . . . . 17 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
101 eqid 2774 . . . . . . . . . . . . . . . . . . . 20 (𝐾s 𝐴) = (𝐾s 𝐴)
102101, 63ressle 16287 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
10318, 102syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
104101, 6ressbas2 16158 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
1055, 104syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = (Base‘(𝐾s 𝐴)))
106105sqxpeqd 5294 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
107103, 106ineq12d 3973 . . . . . . . . . . . . . . . . 17 (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
108100, 107eqtrd 2808 . . . . . . . . . . . . . . . 16 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
109108cnveqd 5448 . . . . . . . . . . . . . . 15 (𝜑( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
110109fveq2d 6352 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
111108fveq2d 6352 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
11298, 110, 1113eqtr4d 2818 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
11392, 112syl5reqr 2823 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
114113eleq2d 2839 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11587, 114raleqbidv 3305 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11679, 115mpbird 248 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
117 ralunb 3952 . . . . . . . . 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)))))
11856, 116, 117sylanbrc 573 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
119 ralunb 3952 . . . . . . . 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)))))
12054, 118, 119sylanbrc 573 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
121 eqid 2774 . . . . . . . 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))
122121fmpt 6540 . . . . . . 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))))
123120, 122sylib 209 . . . . . 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))))
124123frnd 6203 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12537, 124eqsstrd 3795 . . . 4 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
126 tgfiss 21036 . . . 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))))
12730, 125, 126syl2anc 574 . . 3 (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12824, 127eqsstrd 3795 . 2 (𝜑 → ((ordTop‘ ) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
1299, 128eqssd 3775 1 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 383   = wceq 1634  wcel 2148  wral 3064  {crab 3068  Vcvv 3355  cun 3727  cin 3728  wss 3729  {csn 4326   cuni 4585   class class class wbr 4797  cmpt 4876   × cxp 5261  ccnv 5262  dom cdm 5263  ran crn 5264  wf 6038  cfv 6042  (class class class)co 6812  ficfi 8493  Basecbs 16084  s cress 16085  lecple 16176  t crest 16309  topGenctg 16326  ordTopcordt 16387   Preset cpreset 17154  Posetcpo 17168  Tosetctos 17261  ODualcodu 17356  Topctop 20938  TopOnctopon 20955  TopBasesctb 20990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-iin 4668  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-oadd 7738  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-fin 8134  df-fi 8494  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-nn 11244  df-2 11302  df-3 11303  df-4 11304  df-5 11305  df-6 11306  df-7 11307  df-8 11308  df-9 11309  df-dec 11718  df-ndx 16087  df-slot 16088  df-base 16090  df-sets 16091  df-ress 16092  df-ple 16189  df-rest 16311  df-topgen 16332  df-ordt 16389  df-preset 17156  df-poset 17174  df-toset 17262  df-odu 17357  df-top 20939  df-topon 20956  df-bases 20991
This theorem is referenced by: (None)
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