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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ordtrestNEW Structured version   Visualization version   GIF version

Theorem ordtrestNEW 33749
Description: The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (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))
Assertion
Ref Expression
ordtrestNEW ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))

Proof of Theorem ordtrestNEW
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtNEW.l . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2 fvex 6906 . . . . . 6 (le‘𝐾) ∈ V
32inex1 5314 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2822 . . . 4 ∈ V
54inex1 5314 . . 3 ( ∩ (𝐴 × 𝐴)) ∈ V
6 eqid 2726 . . . 4 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
7 eqid 2726 . . . 4 ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
8 eqid 2726 . . . 4 ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
96, 7, 8ordtval 23181 . . 3 (( ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ∩ (𝐴 × 𝐴))) = (topGen‘(fi‘({dom ( ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))))))
105, 9mp1i 13 . 2 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) = (topGen‘(fi‘({dom ( ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))))))
11 ordttop 23192 . . . . . 6 ( ∈ V → (ordTop‘ ) ∈ Top)
124, 11ax-mp 5 . . . . 5 (ordTop‘ ) ∈ Top
13 ordtNEW.b . . . . . . 7 𝐵 = (Base‘𝐾)
14 fvex 6906 . . . . . . 7 (Base‘𝐾) ∈ V
1513, 14eqeltri 2822 . . . . . 6 𝐵 ∈ V
1615ssex 5318 . . . . 5 (𝐴𝐵𝐴 ∈ V)
17 resttop 23152 . . . . 5 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1812, 16, 17sylancr 585 . . . 4 (𝐴𝐵 → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1918adantl 480 . . 3 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
2013ressprs 32836 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Proset )
21 eqid 2726 . . . . . . . . . 10 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
22 eqid 2726 . . . . . . . . . 10 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2321, 22prsdm 33742 . . . . . . . . 9 ((𝐾s 𝐴) ∈ Proset → dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = (Base‘(𝐾s 𝐴)))
2420, 23syl 17 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = (Base‘(𝐾s 𝐴)))
25 eqid 2726 . . . . . . . . . . . . . 14 (𝐾s 𝐴) = (𝐾s 𝐴)
2625, 13ressbas2 17246 . . . . . . . . . . . . 13 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
27 fvex 6906 . . . . . . . . . . . . 13 (Base‘(𝐾s 𝐴)) ∈ V
2826, 27eqeltrdi 2834 . . . . . . . . . . . 12 (𝐴𝐵𝐴 ∈ V)
29 eqid 2726 . . . . . . . . . . . . 13 (le‘𝐾) = (le‘𝐾)
3025, 29ressle 17389 . . . . . . . . . . . 12 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3128, 30syl 17 . . . . . . . . . . 11 (𝐴𝐵 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3231adantl 480 . . . . . . . . . 10 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3326adantl 480 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 = (Base‘(𝐾s 𝐴)))
3433sqxpeqd 5706 . . . . . . . . . 10 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
3532, 34ineq12d 4211 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3635dmeqd 5904 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3724, 36, 333eqtr4d 2776 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
3813, 1prsss 33744 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
3938dmeqd 5904 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
4013, 1prsdm 33742 . . . . . . . . . 10 (𝐾 ∈ Proset → dom = 𝐵)
4140sseq2d 4011 . . . . . . . . 9 (𝐾 ∈ Proset → (𝐴 ⊆ dom 𝐴𝐵))
4241biimpar 476 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 ⊆ dom )
43 sseqin2 4213 . . . . . . . 8 (𝐴 ⊆ dom ↔ (dom 𝐴) = 𝐴)
4442, 43sylib 217 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (dom 𝐴) = 𝐴)
4537, 39, 443eqtr4d 2776 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴))
464, 11mp1i 13 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ Top)
4716adantl 480 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 ∈ V)
48 eqid 2726 . . . . . . . . . 10 dom = dom
4948ordttopon 23185 . . . . . . . . 9 ( ∈ V → (ordTop‘ ) ∈ (TopOn‘dom ))
504, 49mp1i 13 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ (TopOn‘dom ))
51 toponmax 22916 . . . . . . . 8 ((ordTop‘ ) ∈ (TopOn‘dom ) → dom ∈ (ordTop‘ ))
5250, 51syl 17 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ∈ (ordTop‘ ))
53 elrestr 17438 . . . . . . 7 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ∈ (ordTop‘ )) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5446, 47, 52, 53syl3anc 1368 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5545, 54eqeltrd 2826 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ) ↾t 𝐴))
5655snssd 4808 . . . 4 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → {dom ( ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ) ↾t 𝐴))
57 rabeq 3434 . . . . . . . . 9 (dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
5845, 57syl 17 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
5945, 58mpteq12dv 5236 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}))
6059rneqd 5936 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}))
61 inrab2 4306 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥}
62 inss2 4228 . . . . . . . . . . . . . 14 (dom 𝐴) ⊆ 𝐴
63 simpr 483 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦 ∈ (dom 𝐴))
6462, 63sselid 3976 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦𝐴)
65 simpr 483 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥 ∈ (dom 𝐴))
6662, 65sselid 3976 . . . . . . . . . . . . . 14 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥𝐴)
6766adantr 479 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑥𝐴)
68 brinxp 5752 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
6964, 67, 68syl2anc 582 . . . . . . . . . . . 12 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
7069notbid 317 . . . . . . . . . . 11 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑦 𝑥 ↔ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥))
7170rabbidva 3426 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
7261, 71eqtrid 2778 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
734, 11mp1i 13 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → (ordTop‘ ) ∈ Top)
7447adantr 479 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝐴 ∈ V)
75 simpl 481 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐾 ∈ Proset )
76 inss1 4227 . . . . . . . . . . . 12 (dom 𝐴) ⊆ dom
7776sseli 3974 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝐴) → 𝑥 ∈ dom )
7848ordtopn1 23186 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
794, 78mpan 688 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8079adantl 480 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8175, 77, 80syl2an 594 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
82 elrestr 17438 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8373, 74, 81, 82syl3anc 1368 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8472, 83eqeltrrd 2827 . . . . . . . 8 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ) ↾t 𝐴))
8584fmpttd 7121 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
8685frnd 6728 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
8760, 86eqsstrd 4017 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
88 rabeq 3434 . . . . . . . . 9 (dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
8945, 88syl 17 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9045, 89mpteq12dv 5236 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))
9190rneqd 5936 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))
92 inrab2 4306 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦}
93 brinxp 5752 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9467, 64, 93syl2anc 582 . . . . . . . . . . . 12 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9594notbid 317 . . . . . . . . . . 11 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑥 𝑦 ↔ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦))
9695rabbidva 3426 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9792, 96eqtrid 2778 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9848ordtopn2 23187 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
994, 98mpan 688 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10099adantl 480 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10175, 77, 100syl2an 594 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
102 elrestr 17438 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10373, 74, 101, 102syl3anc 1368 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10497, 103eqeltrrd 2827 . . . . . . . 8 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ) ↾t 𝐴))
105104fmpttd 7121 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
106105frnd 6728 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
10791, 106eqsstrd 4017 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
10887, 107unssd 4184 . . . 4 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})) ⊆ ((ordTop‘ ) ↾t 𝐴))
10956, 108unssd 4184 . . 3 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ({dom ( ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))) ⊆ ((ordTop‘ ) ↾t 𝐴))
110 tgfiss 22982 . . 3 ((((ordTop‘ ) ↾t 𝐴) ∈ Top ∧ ({dom ( ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))) ⊆ ((ordTop‘ ) ↾t 𝐴)) → (topGen‘(fi‘({dom ( ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))))) ⊆ ((ordTop‘ ) ↾t 𝐴))
11119, 109, 110syl2anc 582 . 2 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (topGen‘(fi‘({dom ( ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))))) ⊆ ((ordTop‘ ) ↾t 𝐴))
11210, 111eqsstrd 4017 1 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cun 3944  cin 3945  wss 3946  {csn 4623   class class class wbr 5145  cmpt 5228   × cxp 5672  dom cdm 5674  ran crn 5675  cfv 6546  (class class class)co 7416  ficfi 9446  Basecbs 17208  s cress 17237  lecple 17268  t crest 17430  topGenctg 17447  ordTopcordt 17509   Proset cproset 18313  Topctop 22883  TopOnctopon 22900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fi 9447  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-dec 12724  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-ple 17281  df-rest 17432  df-topgen 17453  df-ordt 17511  df-proset 18315  df-top 22884  df-topon 22901  df-bases 22937
This theorem is referenced by:  ordtrest2NEW  33751
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