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

Theorem ordtrestNEW 32502
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 6855 . . . . . 6 (le‘𝐾) ∈ V
32inex1 5274 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2834 . . . 4 ∈ V
54inex1 5274 . . 3 ( ∩ (𝐴 × 𝐴)) ∈ V
6 eqid 2736 . . . 4 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
7 eqid 2736 . . . 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 2736 . . . 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 22540 . . 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 22551 . . . . . 6 ( ∈ V → (ordTop‘ ) ∈ Top)
124, 11ax-mp 5 . . . . 5 (ordTop‘ ) ∈ Top
13 ordtNEW.b . . . . . . 7 𝐵 = (Base‘𝐾)
14 fvex 6855 . . . . . . 7 (Base‘𝐾) ∈ V
1513, 14eqeltri 2834 . . . . . 6 𝐵 ∈ V
1615ssex 5278 . . . . 5 (𝐴𝐵𝐴 ∈ V)
17 resttop 22511 . . . . 5 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1812, 16, 17sylancr 587 . . . 4 (𝐴𝐵 → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1918adantl 482 . . 3 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
2013ressprs 31823 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Proset )
21 eqid 2736 . . . . . . . . . 10 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
22 eqid 2736 . . . . . . . . . 10 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2321, 22prsdm 32495 . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (𝐾s 𝐴) = (𝐾s 𝐴)
2625, 13ressbas2 17120 . . . . . . . . . . . . 13 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
27 fvex 6855 . . . . . . . . . . . . 13 (Base‘(𝐾s 𝐴)) ∈ V
2826, 27eqeltrdi 2846 . . . . . . . . . . . 12 (𝐴𝐵𝐴 ∈ V)
29 eqid 2736 . . . . . . . . . . . . 13 (le‘𝐾) = (le‘𝐾)
3025, 29ressle 17261 . . . . . . . . . . . 12 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3128, 30syl 17 . . . . . . . . . . 11 (𝐴𝐵 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3231adantl 482 . . . . . . . . . 10 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3326adantl 482 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 = (Base‘(𝐾s 𝐴)))
3433sqxpeqd 5665 . . . . . . . . . 10 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
3532, 34ineq12d 4173 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3635dmeqd 5861 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3724, 36, 333eqtr4d 2786 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
3813, 1prsss 32497 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
3938dmeqd 5861 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
4013, 1prsdm 32495 . . . . . . . . . 10 (𝐾 ∈ Proset → dom = 𝐵)
4140sseq2d 3976 . . . . . . . . 9 (𝐾 ∈ Proset → (𝐴 ⊆ dom 𝐴𝐵))
4241biimpar 478 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 ⊆ dom )
43 sseqin2 4175 . . . . . . . 8 (𝐴 ⊆ dom ↔ (dom 𝐴) = 𝐴)
4442, 43sylib 217 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (dom 𝐴) = 𝐴)
4537, 39, 443eqtr4d 2786 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴))
464, 11mp1i 13 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ Top)
4716adantl 482 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 ∈ V)
48 eqid 2736 . . . . . . . . . 10 dom = dom
4948ordttopon 22544 . . . . . . . . 9 ( ∈ V → (ordTop‘ ) ∈ (TopOn‘dom ))
504, 49mp1i 13 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ (TopOn‘dom ))
51 toponmax 22275 . . . . . . . 8 ((ordTop‘ ) ∈ (TopOn‘dom ) → dom ∈ (ordTop‘ ))
5250, 51syl 17 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ∈ (ordTop‘ ))
53 elrestr 17310 . . . . . . 7 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ∈ (ordTop‘ )) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5446, 47, 52, 53syl3anc 1371 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5545, 54eqeltrd 2838 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ) ↾t 𝐴))
5655snssd 4769 . . . 4 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → {dom ( ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ) ↾t 𝐴))
57 rabeq 3421 . . . . . . . . 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 5196 . . . . . . 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 5893 . . . . . 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 4267 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥}
62 inss2 4189 . . . . . . . . . . . . . 14 (dom 𝐴) ⊆ 𝐴
63 simpr 485 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦 ∈ (dom 𝐴))
6462, 63sselid 3942 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦𝐴)
65 simpr 485 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥 ∈ (dom 𝐴))
6662, 65sselid 3942 . . . . . . . . . . . . . 14 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥𝐴)
6766adantr 481 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑥𝐴)
68 brinxp 5710 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
6964, 67, 68syl2anc 584 . . . . . . . . . . . 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 3414 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
7261, 71eqtrid 2788 . . . . . . . . 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 481 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝐴 ∈ V)
75 simpl 483 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐾 ∈ Proset )
76 inss1 4188 . . . . . . . . . . . 12 (dom 𝐴) ⊆ dom
7776sseli 3940 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝐴) → 𝑥 ∈ dom )
7848ordtopn1 22545 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
794, 78mpan 688 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8079adantl 482 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8175, 77, 80syl2an 596 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
82 elrestr 17310 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8373, 74, 81, 82syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8472, 83eqeltrrd 2839 . . . . . . . 8 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ) ↾t 𝐴))
8584fmpttd 7063 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
8685frnd 6676 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
8760, 86eqsstrd 3982 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
88 rabeq 3421 . . . . . . . . 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 5196 . . . . . . 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 5893 . . . . . 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 4267 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦}
93 brinxp 5710 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9467, 64, 93syl2anc 584 . . . . . . . . . . . 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 3414 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9792, 96eqtrid 2788 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9848ordtopn2 22546 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
994, 98mpan 688 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10099adantl 482 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10175, 77, 100syl2an 596 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
102 elrestr 17310 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10373, 74, 101, 102syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10497, 103eqeltrrd 2839 . . . . . . . 8 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ) ↾t 𝐴))
105104fmpttd 7063 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
106105frnd 6676 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
10791, 106eqsstrd 3982 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
10887, 107unssd 4146 . . . 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 4146 . . 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 22341 . . 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 584 . 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 3982 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 396   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  cun 3908  cin 3909  wss 3910  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  cfv 6496  (class class class)co 7357  ficfi 9346  Basecbs 17083  s cress 17112  lecple 17140  t crest 17302  topGenctg 17319  ordTopcordt 17381   Proset cproset 18182  Topctop 22242  TopOnctopon 22259
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-ple 17153  df-rest 17304  df-topgen 17325  df-ordt 17383  df-proset 18184  df-top 22243  df-topon 22260  df-bases 22296
This theorem is referenced by:  ordtrest2NEW  32504
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