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Theorem ordtrest2NEW 33922
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 18465 . . . 4 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3 posprs 18362 . . . 4 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
41, 2, 33syl 18 . . 3 (𝜑𝐾 ∈ Proset )
5 ordtrest2NEW.3 . . 3 (𝜑𝐴𝐵)
6 ordtNEW.b . . . 4 𝐵 = (Base‘𝐾)
7 ordtNEW.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
86, 7ordtrestNEW 33920 . . 3 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
94, 5, 8syl2anc 584 . 2 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
10 eqid 2737 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
11 eqid 2737 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})
126, 7, 10, 11ordtprsval 33917 . . . . . . 7 (𝐾 ∈ Proset → (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 7446 . . . . 5 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
15 fibas 22984 . . . . . 6 (fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases
166fvexi 6920 . . . . . . . 8 𝐵 ∈ V
1716a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
1817, 5ssexd 5324 . . . . . 6 (𝜑𝐴 ∈ V)
19 tgrest 23167 . . . . . 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 587 . . . . 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 2780 . . . 4 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)))
22 firest 17477 . . . . 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 6909 . . . 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, 23eqtr4di 2795 . . 3 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))))
25 fvex 6919 . . . . . . . 8 (le‘𝐾) ∈ V
2625inex1 5317 . . . . . . 7 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
277, 26eqeltri 2837 . . . . . 6 ∈ V
2827inex1 5317 . . . . 5 ( ∩ (𝐴 × 𝐴)) ∈ V
29 ordttop 23208 . . . . 5 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
3028, 29mp1i 13 . . . 4 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
316, 7, 10, 11ordtprsuni 33918 . . . . . . . . 9 (𝐾 ∈ Proset → 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 2842 . . . . . . 7 (𝜑 ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
34 uniexb 7784 . . . . . . 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 234 . . . . . 6 (𝜑 → ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
36 restval 17471 . . . . . 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 584 . . . . 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 4223 . . . . . . . . . . . 12 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
395, 38sylib 218 . . . . . . . . . . 11 (𝜑 → (𝐵𝐴) = 𝐴)
40 eqid 2737 . . . . . . . . . . . . . . 15 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
4140ordttopon 23201 . . . . . . . . . . . . . 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 33916 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
444, 5, 43syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
4544fveq2d 6910 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom ( ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4642, 45eleqtrd 2843 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
47 toponmax 22932 . . . . . . . . . . . 12 ((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4846, 47syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4939, 48eqeltrd 2841 . . . . . . . . . 10 (𝜑 → (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
50 elsni 4643 . . . . . . . . . . . 12 (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
5150ineq1d 4219 . . . . . . . . . . 11 (𝑣 ∈ {𝐵} → (𝑣𝐴) = (𝐵𝐴))
5251eleq1d 2826 . . . . . . . . . 10 (𝑣 ∈ {𝐵} → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5349, 52syl5ibrcom 247 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐵} → (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5453ralrimiv 3145 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
55 ordtrest2NEW.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
566, 7, 1, 5, 55ordtrest2NEWlem 33921 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
57 eqid 2737 . . . . . . . . . . . 12 (ODual‘𝐾) = (ODual‘𝐾)
5857, 6odubas 18336 . . . . . . . . . . 11 𝐵 = (Base‘(ODual‘𝐾))
597cnveqi 5885 . . . . . . . . . . . 12 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
60 cnvin 6164 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
61 cnvxp 6177 . . . . . . . . . . . . . 14 (𝐵 × 𝐵) = (𝐵 × 𝐵)
6261ineq2i 4217 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
63 eqid 2737 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
6457, 63oduleval 18334 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘(ODual‘𝐾))
6564ineq1i 4216 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6660, 62, 653eqtri 2769 . . . . . . . . . . . 12 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6759, 66eqtri 2765 . . . . . . . . . . 11 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6857odutos 32958 . . . . . . . . . . . 12 (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
691, 68syl 17 . . . . . . . . . . 11 (𝜑 → (ODual‘𝐾) ∈ Toset)
70 vex 3484 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
71 vex 3484 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
7270, 71brcnv 5893 . . . . . . . . . . . . . . 15 (𝑦 𝑧𝑧 𝑦)
73 vex 3484 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
7471, 73brcnv 5893 . . . . . . . . . . . . . . 15 (𝑧 𝑥𝑥 𝑧)
7572, 74anbi12ci 629 . . . . . . . . . . . . . 14 ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦))
7675rabbii 3442 . . . . . . . . . . . . 13 {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} = {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)}
7776, 55eqsstrid 4022 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
7877ancom2s 650 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
7958, 67, 69, 5, 78ordtrest2NEWlem 33921 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
80 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
8180, 71brcnv 5893 . . . . . . . . . . . . . . . . 17 (𝑤 𝑧𝑧 𝑤)
8281bicomi 224 . . . . . . . . . . . . . . . 16 (𝑧 𝑤𝑤 𝑧)
8382a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 𝑤𝑤 𝑧))
8483notbid 318 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧 𝑤 ↔ ¬ 𝑤 𝑧))
8584rabbidv 3444 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝐵 ∣ ¬ 𝑧 𝑤} = {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
8685mpteq2dv 5244 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
8786rneqd 5949 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
886ressprs 32954 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Proset )
894, 5, 88syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾s 𝐴) ∈ Proset )
90 eqid 2737 . . . . . . . . . . . . . . . 16 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
91 eqid 2737 . . . . . . . . . . . . . . . 16 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
9290, 91ordtcnvNEW 33919 . . . . . . . . . . . . . . 15 ((𝐾s 𝐴) ∈ Proset → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
9389, 92syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
946, 7prsss 33915 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
954, 5, 94syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
96 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (𝐾s 𝐴) = (𝐾s 𝐴)
9796, 63ressle 17424 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
9818, 97syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
9996, 6ressbas2 17283 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
1005, 99syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = (Base‘(𝐾s 𝐴)))
101100sqxpeqd 5717 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
10298, 101ineq12d 4221 . . . . . . . . . . . . . . . . 17 (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
10395, 102eqtrd 2777 . . . . . . . . . . . . . . . 16 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
104103cnveqd 5886 . . . . . . . . . . . . . . 15 (𝜑( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
105104fveq2d 6910 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
106103fveq2d 6910 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
10793, 105, 1063eqtr4d 2787 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
108 cnvin 6164 . . . . . . . . . . . . . . 15 ( ∩ (𝐴 × 𝐴)) = ( (𝐴 × 𝐴))
109 cnvxp 6177 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
110109ineq2i 4217 . . . . . . . . . . . . . . 15 ( (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
111108, 110eqtri 2765 . . . . . . . . . . . . . 14 ( ∩ (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
112111fveq2i 6909 . . . . . . . . . . . . 13 (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴)))
113107, 112eqtr3di 2792 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
114113eleq2d 2827 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11587, 114raleqbidv 3346 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11679, 115mpbird 257 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
117 ralunb 4197 . . . . . . . . 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 583 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
119 ralunb 4197 . . . . . . . 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 583 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
121 eqid 2737 . . . . . . . 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 7130 . . . . . . 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 218 . . . . . 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 6744 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12537, 124eqsstrd 4018 . . . 4 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
126 tgfiss 22998 . . . 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 584 . . 3 (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12824, 127eqsstrd 4018 . 2 (𝜑 → ((ordTop‘ ) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
1299, 128eqssd 4001 1 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cun 3949  cin 3950  wss 3951  {csn 4626   cuni 4907   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  ficfi 9450  Basecbs 17247  s cress 17274  lecple 17304  t crest 17465  topGenctg 17482  ordTopcordt 17544  ODualcodu 18331   Proset cproset 18338  Posetcpo 18353  Tosetctos 18461  Topctop 22899  TopOnctopon 22916  TopBasesctb 22952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-ple 17317  df-rest 17467  df-topgen 17488  df-ordt 17546  df-odu 18332  df-proset 18340  df-poset 18359  df-toset 18462  df-top 22900  df-topon 22917  df-bases 22953
This theorem is referenced by: (None)
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