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Theorem ordtrest2NEW 31191
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 30649 . . . 4 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3 posprs 17555 . . . 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 31189 . . 3 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
94, 5, 8syl2anc 587 . 2 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
10 eqid 2824 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
11 eqid 2824 . . . . . . . 8 ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})
126, 7, 10, 11ordtprsval 31186 . . . . . . 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 7160 . . . . 5 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))))) ↾t 𝐴))
15 fibas 21578 . . . . . 6 (fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ∈ TopBases
166fvexi 6672 . . . . . . . 8 𝐵 ∈ V
1716a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
1817, 5ssexd 5214 . . . . . 6 (𝜑𝐴 ∈ V)
19 tgrest 21760 . . . . . 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 590 . . . . 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 2862 . . . 4 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))) ↾t 𝐴)))
22 firest 16702 . . . . 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 6661 . . . 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 2877 . . 3 (𝜑 → ((ordTop‘ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))))
25 fvex 6671 . . . . . . . 8 (le‘𝐾) ∈ V
2625inex1 5207 . . . . . . 7 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
277, 26eqeltri 2912 . . . . . 6 ∈ V
2827inex1 5207 . . . . 5 ( ∩ (𝐴 × 𝐴)) ∈ V
29 ordttop 21801 . . . . 5 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
3028, 29mp1i 13 . . . 4 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ Top)
316, 7, 10, 11ordtprsuni 31187 . . . . . . . . 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 2917 . . . . . . 7 (𝜑 ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
34 uniexb 7476 . . . . . . 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 237 . . . . . 6 (𝜑 → ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ∈ V)
36 restval 16696 . . . . . 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 587 . . . . 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 4176 . . . . . . . . . . . 12 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
395, 38sylib 221 . . . . . . . . . . 11 (𝜑 → (𝐵𝐴) = 𝐴)
40 eqid 2824 . . . . . . . . . . . . . . 15 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
4140ordttopon 21794 . . . . . . . . . . . . . 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 31185 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
444, 5, 43syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
4544fveq2d 6662 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom ( ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4642, 45eleqtrd 2918 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
47 toponmax 21527 . . . . . . . . . . . 12 ((ordTop‘( ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4846, 47syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
4939, 48eqeltrd 2916 . . . . . . . . . 10 (𝜑 → (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
50 elsni 4566 . . . . . . . . . . . 12 (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
5150ineq1d 4172 . . . . . . . . . . 11 (𝑣 ∈ {𝐵} → (𝑣𝐴) = (𝐵𝐴))
5251eleq1d 2900 . . . . . . . . . 10 (𝑣 ∈ {𝐵} → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝐵𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5349, 52syl5ibrcom 250 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝐵} → (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
5453ralrimiv 3176 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
55 ordtrest2NEW.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)
566, 7, 1, 5, 55ordtrest2NEWlem 31190 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
57 eqid 2824 . . . . . . . . . . . 12 (ODual‘𝐾) = (ODual‘𝐾)
5857, 6odubas 17739 . . . . . . . . . . 11 𝐵 = (Base‘(ODual‘𝐾))
597cnveqi 5732 . . . . . . . . . . . 12 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
60 cnvin 5990 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
61 cnvxp 6001 . . . . . . . . . . . . . 14 (𝐵 × 𝐵) = (𝐵 × 𝐵)
6261ineq2i 4170 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
63 eqid 2824 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
6457, 63oduleval 17737 . . . . . . . . . . . . . 14 (le‘𝐾) = (le‘(ODual‘𝐾))
6564ineq1i 4169 . . . . . . . . . . . . 13 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6660, 62, 653eqtri 2851 . . . . . . . . . . . 12 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6759, 66eqtri 2847 . . . . . . . . . . 11 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
6857odutos 30654 . . . . . . . . . . . 12 (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
691, 68syl 17 . . . . . . . . . . 11 (𝜑 → (ODual‘𝐾) ∈ Toset)
70 vex 3483 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
71 vex 3483 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
7270, 71brcnv 5740 . . . . . . . . . . . . . . 15 (𝑦 𝑧𝑧 𝑦)
73 vex 3483 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
7471, 73brcnv 5740 . . . . . . . . . . . . . . 15 (𝑧 𝑥𝑥 𝑧)
7572, 74anbi12ci 630 . . . . . . . . . . . . . 14 ((𝑦 𝑧𝑧 𝑥) ↔ (𝑥 𝑧𝑧 𝑦))
7675rabbii 3459 . . . . . . . . . . . . 13 {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} = {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)}
7776, 55eqsstrid 4000 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
7877ancom2s 649 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧𝐵 ∣ (𝑦 𝑧𝑧 𝑥)} ⊆ 𝐴)
7958, 67, 69, 5, 78ordtrest2NEWlem 31190 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
80 vex 3483 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
8180, 71brcnv 5740 . . . . . . . . . . . . . . . . 17 (𝑤 𝑧𝑧 𝑤)
8281bicomi 227 . . . . . . . . . . . . . . . 16 (𝑧 𝑤𝑤 𝑧)
8382a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 𝑤𝑤 𝑧))
8483notbid 321 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧 𝑤 ↔ ¬ 𝑤 𝑧))
8584rabbidv 3466 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝐵 ∣ ¬ 𝑧 𝑤} = {𝑤𝐵 ∣ ¬ 𝑤 𝑧})
8685mpteq2dv 5148 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
8786rneqd 5795 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}) = ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}))
88 cnvin 5990 . . . . . . . . . . . . . . 15 ( ∩ (𝐴 × 𝐴)) = ( (𝐴 × 𝐴))
89 cnvxp 6001 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
9089ineq2i 4170 . . . . . . . . . . . . . . 15 ( (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9188, 90eqtri 2847 . . . . . . . . . . . . . 14 ( ∩ (𝐴 × 𝐴)) = ( ∩ (𝐴 × 𝐴))
9291fveq2i 6661 . . . . . . . . . . . . 13 (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴)))
936ressprs 30646 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Proset )
944, 5, 93syl2anc 587 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾s 𝐴) ∈ Proset )
95 eqid 2824 . . . . . . . . . . . . . . . 16 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
96 eqid 2824 . . . . . . . . . . . . . . . 16 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
9795, 96ordtcnvNEW 31188 . . . . . . . . . . . . . . 15 ((𝐾s 𝐴) ∈ Proset → (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 31184 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
1004, 5, 99syl2anc 587 . . . . . . . . . . . . . . . . 17 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
101 eqid 2824 . . . . . . . . . . . . . . . . . . . 20 (𝐾s 𝐴) = (𝐾s 𝐴)
102101, 63ressle 16668 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
10318, 102syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
104101, 6ressbas2 16551 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
1055, 104syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 = (Base‘(𝐾s 𝐴)))
106105sqxpeqd 5574 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
107103, 106ineq12d 4174 . . . . . . . . . . . . . . . . 17 (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
108100, 107eqtrd 2859 . . . . . . . . . . . . . . . 16 (𝜑 → ( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
109108cnveqd 5733 . . . . . . . . . . . . . . 15 (𝜑( ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
110109fveq2d 6662 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
111108fveq2d 6662 . . . . . . . . . . . . . 14 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
11298, 110, 1113eqtr4d 2869 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
11392, 112syl5reqr 2874 . . . . . . . . . . . 12 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = (ordTop‘( ∩ (𝐴 × 𝐴))))
114113eleq2d 2901 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11587, 114raleqbidv 3393 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴)))))
11679, 115mpbird 260 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
117 ralunb 4152 . . . . . . . . 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 586 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
119 ralunb 4152 . . . . . . . 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 586 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤})))(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
121 eqid 2824 . . . . . . . 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 6862 . . . . . . 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 221 . . . . . 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 6509 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12537, 124eqsstrd 3990 . . . 4 (𝜑 → (({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
126 tgfiss 21592 . . . 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 587 . . 3 (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧}) ∪ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑧 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
12824, 127eqsstrd 3990 . 2 (𝜑 → ((ordTop‘ ) ↾t 𝐴) ⊆ (ordTop‘( ∩ (𝐴 × 𝐴))))
1299, 128eqssd 3969 1 (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  cun 3917  cin 3918  wss 3919  {csn 4549   cuni 4824   class class class wbr 5052  cmpt 5132   × cxp 5540  ccnv 5541  dom cdm 5542  ran crn 5543  wf 6339  cfv 6343  (class class class)co 7145  ficfi 8865  Basecbs 16479  s cress 16480  lecple 16568  t crest 16690  topGenctg 16707  ordTopcordt 16768   Proset cproset 17532  Posetcpo 17546  Tosetctos 17639  ODualcodu 17734  Topctop 21494  TopOnctopon 21511  TopBasesctb 21546
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-fi 8866  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-5 11696  df-6 11697  df-7 11698  df-8 11699  df-9 11700  df-dec 12092  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-ple 16581  df-rest 16692  df-topgen 16713  df-ordt 16770  df-proset 17534  df-poset 17552  df-toset 17640  df-odu 17735  df-top 21495  df-topon 21512  df-bases 21547
This theorem is referenced by: (None)
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