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

Theorem ordtrestNEW 30276
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 ∈ Preset ∧ 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 6362 . . . . . 6 (le‘𝐾) ∈ V
32inex1 4951 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2835 . . . 4 ∈ V
54inex1 4951 . . 3 ( ∩ (𝐴 × 𝐴)) ∈ V
6 eqid 2760 . . . 4 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
7 eqid 2760 . . . 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 2760 . . . 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 21195 . . 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 ∈ Preset ∧ 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 21206 . . . . . 6 ( ∈ V → (ordTop‘ ) ∈ Top)
124, 11ax-mp 5 . . . . 5 (ordTop‘ ) ∈ Top
13 ordtNEW.b . . . . . . 7 𝐵 = (Base‘𝐾)
14 fvex 6362 . . . . . . 7 (Base‘𝐾) ∈ V
1513, 14eqeltri 2835 . . . . . 6 𝐵 ∈ V
1615ssex 4954 . . . . 5 (𝐴𝐵𝐴 ∈ V)
17 resttop 21166 . . . . 5 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1812, 16, 17sylancr 698 . . . 4 (𝐴𝐵 → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1918adantl 473 . . 3 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
2013ressprs 29964 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )
21 eqid 2760 . . . . . . . . . 10 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
22 eqid 2760 . . . . . . . . . 10 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2321, 22prsdm 30269 . . . . . . . . 9 ((𝐾s 𝐴) ∈ Preset → dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = (Base‘(𝐾s 𝐴)))
2420, 23syl 17 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = (Base‘(𝐾s 𝐴)))
25 eqid 2760 . . . . . . . . . . . . . 14 (𝐾s 𝐴) = (𝐾s 𝐴)
2625, 13ressbas2 16133 . . . . . . . . . . . . 13 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
27 fvex 6362 . . . . . . . . . . . . 13 (Base‘(𝐾s 𝐴)) ∈ V
2826, 27syl6eqel 2847 . . . . . . . . . . . 12 (𝐴𝐵𝐴 ∈ V)
29 eqid 2760 . . . . . . . . . . . . 13 (le‘𝐾) = (le‘𝐾)
3025, 29ressle 16261 . . . . . . . . . . . 12 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3128, 30syl 17 . . . . . . . . . . 11 (𝐴𝐵 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3231adantl 473 . . . . . . . . . 10 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3326adantl 473 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐴 = (Base‘(𝐾s 𝐴)))
3433sqxpeqd 5298 . . . . . . . . . 10 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
3532, 34ineq12d 3958 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3635dmeqd 5481 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3724, 36, 333eqtr4d 2804 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
3813, 1prsss 30271 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
3938dmeqd 5481 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
4013, 1prsdm 30269 . . . . . . . . . 10 (𝐾 ∈ Preset → dom = 𝐵)
4140sseq2d 3774 . . . . . . . . 9 (𝐾 ∈ Preset → (𝐴 ⊆ dom 𝐴𝐵))
4241biimpar 503 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐴 ⊆ dom )
43 sseqin2 3960 . . . . . . . 8 (𝐴 ⊆ dom ↔ (dom 𝐴) = 𝐴)
4442, 43sylib 208 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (dom 𝐴) = 𝐴)
4537, 39, 443eqtr4d 2804 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴))
464, 11mp1i 13 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ Top)
4716adantl 473 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐴 ∈ V)
48 eqid 2760 . . . . . . . . . 10 dom = dom
4948ordttopon 21199 . . . . . . . . 9 ( ∈ V → (ordTop‘ ) ∈ (TopOn‘dom ))
504, 49mp1i 13 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ (TopOn‘dom ))
51 toponmax 20932 . . . . . . . 8 ((ordTop‘ ) ∈ (TopOn‘dom ) → dom ∈ (ordTop‘ ))
5250, 51syl 17 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ∈ (ordTop‘ ))
53 elrestr 16291 . . . . . . 7 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ∈ (ordTop‘ )) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5446, 47, 52, 53syl3anc 1477 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5545, 54eqeltrd 2839 . . . . 5 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ) ↾t 𝐴))
5655snssd 4485 . . . 4 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → {dom ( ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ) ↾t 𝐴))
57 rabeq 3332 . . . . . . . . 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 ∈ Preset ∧ 𝐴𝐵) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
5945, 58mpteq12dv 4885 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}))
6059rneqd 5508 . . . . . 6 ((𝐾 ∈ Preset ∧ 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 4043 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥}
62 inss2 3977 . . . . . . . . . . . . . 14 (dom 𝐴) ⊆ 𝐴
63 simpr 479 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦 ∈ (dom 𝐴))
6462, 63sseldi 3742 . . . . . . . . . . . . 13 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦𝐴)
65 simpr 479 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥 ∈ (dom 𝐴))
6662, 65sseldi 3742 . . . . . . . . . . . . . 14 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥𝐴)
6766adantr 472 . . . . . . . . . . . . 13 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑥𝐴)
68 brinxp 5338 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
6964, 67, 68syl2anc 696 . . . . . . . . . . . 12 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
7069notbid 307 . . . . . . . . . . 11 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑦 𝑥 ↔ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥))
7170rabbidva 3328 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
7261, 71syl5eq 2806 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
734, 11mp1i 13 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → (ordTop‘ ) ∈ Top)
7447adantr 472 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝐴 ∈ V)
75 simpl 474 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐾 ∈ Preset )
76 inss1 3976 . . . . . . . . . . . 12 (dom 𝐴) ⊆ dom
7776sseli 3740 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝐴) → 𝑥 ∈ dom )
7848ordtopn1 21200 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
794, 78mpan 708 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8079adantl 473 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8175, 77, 80syl2an 495 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
82 elrestr 16291 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8373, 74, 81, 82syl3anc 1477 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8472, 83eqeltrrd 2840 . . . . . . . 8 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ) ↾t 𝐴))
85 eqid 2760 . . . . . . . 8 (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
8684, 85fmptd 6548 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
87 frn 6214 . . . . . . 7 ((𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
8886, 87syl 17 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
8960, 88eqsstrd 3780 . . . . 5 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
90 rabeq 3332 . . . . . . . . 9 (dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9145, 90syl 17 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9245, 91mpteq12dv 4885 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}))
9392rneqd 5508 . . . . . 6 ((𝐾 ∈ Preset ∧ 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}))
94 inrab2 4043 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦}
95 brinxp 5338 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9667, 64, 95syl2anc 696 . . . . . . . . . . . 12 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9796notbid 307 . . . . . . . . . . 11 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑥 𝑦 ↔ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦))
9897rabbidva 3328 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9994, 98syl5eq 2806 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
10048ordtopn2 21201 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
1014, 100mpan 708 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
102101adantl 473 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10375, 77, 102syl2an 495 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
104 elrestr 16291 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10573, 74, 103, 104syl3anc 1477 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10699, 105eqeltrrd 2840 . . . . . . . 8 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ) ↾t 𝐴))
107 eqid 2760 . . . . . . . 8 (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
108106, 107fmptd 6548 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
109 frn 6214 . . . . . . 7 ((𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
110108, 109syl 17 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
11193, 110eqsstrd 3780 . . . . 5 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
11289, 111unssd 3932 . . . 4 ((𝐾 ∈ Preset ∧ 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))
11356, 112unssd 3932 . . 3 ((𝐾 ∈ Preset ∧ 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))
114 tgfiss 20997 . . 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))
11519, 113, 114syl2anc 696 . 2 ((𝐾 ∈ Preset ∧ 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))
11610, 115eqsstrd 3780 1 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cun 3713  cin 3714  wss 3715  {csn 4321   class class class wbr 4804  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6813  ficfi 8481  Basecbs 16059  s cress 16060  lecple 16150  t crest 16283  topGenctg 16300  ordTopcordt 16361   Preset cpreset 17127  Topctop 20900  TopOnctopon 20917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-dec 11686  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-ple 16163  df-rest 16285  df-topgen 16306  df-ordt 16363  df-preset 17129  df-top 20901  df-topon 20918  df-bases 20952
This theorem is referenced by:  ordtrest2NEW  30278
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