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

Theorem ordtrestNEW 33882
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 6920 . . . . . 6 (le‘𝐾) ∈ V
32inex1 5323 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2835 . . . 4 ∈ V
54inex1 5323 . . 3 ( ∩ (𝐴 × 𝐴)) ∈ V
6 eqid 2735 . . . 4 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
7 eqid 2735 . . . 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 2735 . . . 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 23213 . . 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 23224 . . . . . 6 ( ∈ V → (ordTop‘ ) ∈ Top)
124, 11ax-mp 5 . . . . 5 (ordTop‘ ) ∈ Top
13 ordtNEW.b . . . . . . 7 𝐵 = (Base‘𝐾)
14 fvex 6920 . . . . . . 7 (Base‘𝐾) ∈ V
1513, 14eqeltri 2835 . . . . . 6 𝐵 ∈ V
1615ssex 5327 . . . . 5 (𝐴𝐵𝐴 ∈ V)
17 resttop 23184 . . . . 5 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1812, 16, 17sylancr 587 . . . 4 (𝐴𝐵 → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1918adantl 481 . . 3 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
2013ressprs 32939 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Proset )
21 eqid 2735 . . . . . . . . . 10 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
22 eqid 2735 . . . . . . . . . 10 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2321, 22prsdm 33875 . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (𝐾s 𝐴) = (𝐾s 𝐴)
2625, 13ressbas2 17283 . . . . . . . . . . . . 13 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
27 fvex 6920 . . . . . . . . . . . . 13 (Base‘(𝐾s 𝐴)) ∈ V
2826, 27eqeltrdi 2847 . . . . . . . . . . . 12 (𝐴𝐵𝐴 ∈ V)
29 eqid 2735 . . . . . . . . . . . . 13 (le‘𝐾) = (le‘𝐾)
3025, 29ressle 17426 . . . . . . . . . . . 12 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3128, 30syl 17 . . . . . . . . . . 11 (𝐴𝐵 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3231adantl 481 . . . . . . . . . 10 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3326adantl 481 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 = (Base‘(𝐾s 𝐴)))
3433sqxpeqd 5721 . . . . . . . . . 10 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
3532, 34ineq12d 4229 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3635dmeqd 5919 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3724, 36, 333eqtr4d 2785 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
3813, 1prsss 33877 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
3938dmeqd 5919 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
4013, 1prsdm 33875 . . . . . . . . . 10 (𝐾 ∈ Proset → dom = 𝐵)
4140sseq2d 4028 . . . . . . . . 9 (𝐾 ∈ Proset → (𝐴 ⊆ dom 𝐴𝐵))
4241biimpar 477 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 ⊆ dom )
43 sseqin2 4231 . . . . . . . 8 (𝐴 ⊆ dom ↔ (dom 𝐴) = 𝐴)
4442, 43sylib 218 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (dom 𝐴) = 𝐴)
4537, 39, 443eqtr4d 2785 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴))
464, 11mp1i 13 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ Top)
4716adantl 481 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐴 ∈ V)
48 eqid 2735 . . . . . . . . . 10 dom = dom
4948ordttopon 23217 . . . . . . . . 9 ( ∈ V → (ordTop‘ ) ∈ (TopOn‘dom ))
504, 49mp1i 13 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ (TopOn‘dom ))
51 toponmax 22948 . . . . . . . 8 ((ordTop‘ ) ∈ (TopOn‘dom ) → dom ∈ (ordTop‘ ))
5250, 51syl 17 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ∈ (ordTop‘ ))
53 elrestr 17475 . . . . . . 7 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ∈ (ordTop‘ )) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5446, 47, 52, 53syl3anc 1370 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5545, 54eqeltrd 2839 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ) ↾t 𝐴))
5655snssd 4814 . . . 4 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → {dom ( ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ) ↾t 𝐴))
57 rabeq 3448 . . . . . . . . 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 5239 . . . . . . 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 5952 . . . . . 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 4323 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥}
62 inss2 4246 . . . . . . . . . . . . . 14 (dom 𝐴) ⊆ 𝐴
63 simpr 484 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦 ∈ (dom 𝐴))
6462, 63sselid 3993 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦𝐴)
65 simpr 484 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥 ∈ (dom 𝐴))
6662, 65sselid 3993 . . . . . . . . . . . . . 14 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥𝐴)
6766adantr 480 . . . . . . . . . . . . 13 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑥𝐴)
68 brinxp 5767 . . . . . . . . . . . . 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 318 . . . . . . . . . . 11 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑦 𝑥 ↔ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥))
7170rabbidva 3440 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
7261, 71eqtrid 2787 . . . . . . . . 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 480 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝐴 ∈ V)
75 simpl 482 . . . . . . . . . . 11 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → 𝐾 ∈ Proset )
76 inss1 4245 . . . . . . . . . . . 12 (dom 𝐴) ⊆ dom
7776sseli 3991 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝐴) → 𝑥 ∈ dom )
7848ordtopn1 23218 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
794, 78mpan 690 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8079adantl 481 . . . . . . . . . . 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 17475 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8373, 74, 81, 82syl3anc 1370 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8472, 83eqeltrrd 2840 . . . . . . . 8 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ) ↾t 𝐴))
8584fmpttd 7135 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
8685frnd 6745 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
8760, 86eqsstrd 4034 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
88 rabeq 3448 . . . . . . . . 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 5239 . . . . . . 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 5952 . . . . . 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 4323 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦}
93 brinxp 5767 . . . . . . . . . . . . 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 318 . . . . . . . . . . 11 ((((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑥 𝑦 ↔ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦))
9695rabbidva 3440 . . . . . . . . . 10 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9792, 96eqtrid 2787 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9848ordtopn2 23219 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
994, 98mpan 690 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10099adantl 481 . . . . . . . . . . 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 17475 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10373, 74, 101, 102syl3anc 1370 . . . . . . . . 9 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10497, 103eqeltrrd 2840 . . . . . . . 8 (((𝐾 ∈ Proset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ) ↾t 𝐴))
105104fmpttd 7135 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
106105frnd 6745 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
10791, 106eqsstrd 4034 . . . . 5 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
10887, 107unssd 4202 . . . 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 4202 . . 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 23014 . . 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 4034 1 ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cun 3961  cin 3962  wss 3963  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cfv 6563  (class class class)co 7431  ficfi 9448  Basecbs 17245  s cress 17274  lecple 17305  t crest 17467  topGenctg 17484  ordTopcordt 17546   Proset cproset 18350  Topctop 22915  TopOnctopon 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-ple 17318  df-rest 17469  df-topgen 17490  df-ordt 17548  df-proset 18352  df-top 22916  df-topon 22933  df-bases 22969
This theorem is referenced by:  ordtrest2NEW  33884
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