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Theorem ordtrestNEW 29746
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 6158 . . . . . 6 (le‘𝐾) ∈ V
32inex1 4759 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2694 . . . 4 ∈ V
54inex1 4759 . . 3 ( ∩ (𝐴 × 𝐴)) ∈ V
6 eqid 2621 . . . 4 dom ( ∩ (𝐴 × 𝐴)) = dom ( ∩ (𝐴 × 𝐴))
7 eqid 2621 . . . 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 2621 . . . 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 20903 . . 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 20914 . . . . . 6 ( ∈ V → (ordTop‘ ) ∈ Top)
124, 11ax-mp 5 . . . . 5 (ordTop‘ ) ∈ Top
13 ordtNEW.b . . . . . . 7 𝐵 = (Base‘𝐾)
14 fvex 6158 . . . . . . 7 (Base‘𝐾) ∈ V
1513, 14eqeltri 2694 . . . . . 6 𝐵 ∈ V
1615ssex 4762 . . . . 5 (𝐴𝐵𝐴 ∈ V)
17 resttop 20874 . . . . 5 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1812, 16, 17sylancr 694 . . . 4 (𝐴𝐵 → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
1918adantl 482 . . 3 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ((ordTop‘ ) ↾t 𝐴) ∈ Top)
2013ressprs 29437 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )
21 eqid 2621 . . . . . . . . . 10 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
22 eqid 2621 . . . . . . . . . 10 ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2321, 22prsdm 29739 . . . . . . . . 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 2621 . . . . . . . . . . . . . 14 (𝐾s 𝐴) = (𝐾s 𝐴)
2625, 13ressbas2 15852 . . . . . . . . . . . . 13 (𝐴𝐵𝐴 = (Base‘(𝐾s 𝐴)))
27 fvex 6158 . . . . . . . . . . . . 13 (Base‘(𝐾s 𝐴)) ∈ V
2826, 27syl6eqel 2706 . . . . . . . . . . . 12 (𝐴𝐵𝐴 ∈ V)
29 eqid 2621 . . . . . . . . . . . . 13 (le‘𝐾) = (le‘𝐾)
3025, 29ressle 15980 . . . . . . . . . . . 12 (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3128, 30syl 17 . . . . . . . . . . 11 (𝐴𝐵 → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3231adantl 482 . . . . . . . . . 10 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (le‘𝐾) = (le‘(𝐾s 𝐴)))
3326adantl 482 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐴 = (Base‘(𝐾s 𝐴)))
3433sqxpeqd 5101 . . . . . . . . . 10 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
3532, 34ineq12d 3793 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3635dmeqd 5286 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾s 𝐴)) ∩ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
3724, 36, 333eqtr4d 2665 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
3813, 1prsss 29741 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
3938dmeqd 5286 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
4013, 1prsdm 29739 . . . . . . . . . 10 (𝐾 ∈ Preset → dom = 𝐵)
4140sseq2d 3612 . . . . . . . . 9 (𝐾 ∈ Preset → (𝐴 ⊆ dom 𝐴𝐵))
4241biimpar 502 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐴 ⊆ dom )
43 sseqin2 3795 . . . . . . . 8 (𝐴 ⊆ dom ↔ (dom 𝐴) = 𝐴)
4442, 43sylib 208 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (dom 𝐴) = 𝐴)
4537, 39, 443eqtr4d 2665 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = (dom 𝐴))
464, 11mp1i 13 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ Top)
4716adantl 482 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐴 ∈ V)
48 eqid 2621 . . . . . . . . . 10 dom = dom
4948ordttopon 20907 . . . . . . . . 9 ( ∈ V → (ordTop‘ ) ∈ (TopOn‘dom ))
504, 49mp1i 13 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (ordTop‘ ) ∈ (TopOn‘dom ))
51 toponmax 20643 . . . . . . . 8 ((ordTop‘ ) ∈ (TopOn‘dom ) → dom ∈ (ordTop‘ ))
5250, 51syl 17 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ∈ (ordTop‘ ))
53 elrestr 16010 . . . . . . 7 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ∈ (ordTop‘ )) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5446, 47, 52, 53syl3anc 1323 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (dom 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
5545, 54eqeltrd 2698 . . . . 5 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ) ↾t 𝐴))
5655snssd 4309 . . . 4 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → {dom ( ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ) ↾t 𝐴))
57 rabeq 3179 . . . . . . . . 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 4693 . . . . . . 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 5313 . . . . . 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 3876 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥}
62 inss2 3812 . . . . . . . . . . . . . 14 (dom 𝐴) ⊆ 𝐴
63 simpr 477 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦 ∈ (dom 𝐴))
6462, 63sseldi 3581 . . . . . . . . . . . . 13 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑦𝐴)
65 simpr 477 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥 ∈ (dom 𝐴))
6662, 65sseldi 3581 . . . . . . . . . . . . . 14 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝑥𝐴)
6766adantr 481 . . . . . . . . . . . . 13 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → 𝑥𝐴)
68 brinxp 5142 . . . . . . . . . . . . 13 ((𝑦𝐴𝑥𝐴) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
6964, 67, 68syl2anc 692 . . . . . . . . . . . 12 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (𝑦 𝑥𝑦( ∩ (𝐴 × 𝐴))𝑥))
7069notbid 308 . . . . . . . . . . 11 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑦 𝑥 ↔ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥))
7170rabbidva 3176 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦 𝑥} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
7261, 71syl5eq 2667 . . . . . . . . 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 481 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → 𝐴 ∈ V)
75 simpl 473 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → 𝐾 ∈ Preset )
76 inss1 3811 . . . . . . . . . . . 12 (dom 𝐴) ⊆ dom
7776sseli 3579 . . . . . . . . . . 11 (𝑥 ∈ (dom 𝐴) → 𝑥 ∈ dom )
7848ordtopn1 20908 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
794, 78mpan 705 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8079adantl 482 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
8175, 77, 80syl2an 494 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ ))
82 elrestr 16010 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8373, 74, 81, 82syl3anc 1323 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
8472, 83eqeltrrd 2699 . . . . . . . 8 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ) ↾t 𝐴))
85 eqid 2621 . . . . . . . 8 (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥})
8684, 85fmptd 6340 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
87 frn 6010 . . . . . . 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 3618 . . . . 5 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ) ↾t 𝐴))
90 rabeq 3179 . . . . . . . . 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 4693 . . . . . . 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 5313 . . . . . 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 3876 . . . . . . . . . 10 ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦}
95 brinxp 5142 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9667, 64, 95syl2anc 692 . . . . . . . . . . . 12 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (𝑥 𝑦𝑥( ∩ (𝐴 × 𝐴))𝑦))
9796notbid 308 . . . . . . . . . . 11 ((((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) ∧ 𝑦 ∈ (dom 𝐴)) → (¬ 𝑥 𝑦 ↔ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦))
9897rabbidva 3176 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥 𝑦} = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
9994, 98syl5eq 2667 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
10048ordtopn2 20909 . . . . . . . . . . . . 13 (( ∈ V ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
1014, 100mpan 705 . . . . . . . . . . . 12 (𝑥 ∈ dom → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
102101adantl 482 . . . . . . . . . . 11 ((𝐾 ∈ Preset ∧ 𝑥 ∈ dom ) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
10375, 77, 102syl2an 494 . . . . . . . . . 10 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ ))
104 elrestr 16010 . . . . . . . . . 10 (((ordTop‘ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∈ (ordTop‘ )) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10573, 74, 103, 104syl3anc 1323 . . . . . . . . 9 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → ({𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ) ↾t 𝐴))
10699, 105eqeltrrd 2699 . . . . . . . 8 (((𝐾 ∈ Preset ∧ 𝐴𝐵) ∧ 𝑥 ∈ (dom 𝐴)) → {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ) ↾t 𝐴))
107 eqid 2621 . . . . . . . 8 (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦})
108106, 107fmptd 6340 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝑥 ∈ (dom 𝐴) ↦ {𝑦 ∈ (dom 𝐴) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}):(dom 𝐴)⟶((ordTop‘ ) ↾t 𝐴))
109 frn 6010 . . . . . . 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 3618 . . . . 5 ((𝐾 ∈ Preset ∧ 𝐴𝐵) → ran (𝑥 ∈ dom ( ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ) ↾t 𝐴))
11289, 111unssd 3767 . . . 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 3767 . . 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 20706 . . 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 692 . 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 3618 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 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cun 3553  cin 3554  wss 3555  {csn 4148   class class class wbr 4613  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  ficfi 8260  Basecbs 15781  s cress 15782  lecple 15869  t crest 16002  topGenctg 16019  ordTopcordt 16080   Preset cpreset 16847  Topctop 20617  TopOnctopon 20618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-dec 11438  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-ple 15882  df-rest 16004  df-topgen 16025  df-ordt 16082  df-preset 16849  df-top 20621  df-bases 20622  df-topon 20623
This theorem is referenced by:  ordtrest2NEW  29748
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