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Theorem ushrisomgr 43838
Description: A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.)
Hypotheses
Ref Expression
ushrisomgr.v 𝑉 = (Vtx‘𝐺)
ushrisomgr.e 𝐸 = (Edg‘𝐺)
ushrisomgr.s 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩
Assertion
Ref Expression
ushrisomgr (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻)

Proof of Theorem ushrisomgr
Dummy variables 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ushrisomgr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
21fvexi 6681 . . . . 5 𝑉 ∈ V
32a1i 11 . . . 4 (𝐺 ∈ USHGraph → 𝑉 ∈ V)
43resiexd 6974 . . 3 (𝐺 ∈ USHGraph → ( I ↾ 𝑉) ∈ V)
5 f1oi 6649 . . . . . 6 ( I ↾ 𝑉):𝑉1-1-onto𝑉
65a1i 11 . . . . 5 (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉1-1-onto𝑉)
7 ushrisomgr.s . . . . . . . 8 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩
87fveq2i 6670 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
9 ushrisomgr.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
109fvexi 6681 . . . . . . . . . 10 𝐸 ∈ V
11 id 22 . . . . . . . . . . 11 (𝐸 ∈ V → 𝐸 ∈ V)
1211resiexd 6974 . . . . . . . . . 10 (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V)
1310, 12ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐸) ∈ V
142, 13pm3.2i 471 . . . . . . . 8 (𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V)
15 opvtxfv 26703 . . . . . . . 8 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
1614, 15mp1i 13 . . . . . . 7 (𝐺 ∈ USHGraph → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
178, 16syl5eq 2873 . . . . . 6 (𝐺 ∈ USHGraph → (Vtx‘𝐻) = 𝑉)
1817f1oeq3d 6609 . . . . 5 (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto𝑉))
196, 18mpbird 258 . . . 4 (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻))
20 fvexd 6682 . . . . 5 (𝐺 ∈ USHGraph → (iEdg‘𝐺) ∈ V)
21 eqid 2826 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
221, 21ushgrf 26762 . . . . . . . 8 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}))
23 f1f1orn 6623 . . . . . . . 8 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
2422, 23syl 17 . . . . . . 7 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
257fveq2i 6670 . . . . . . . . . . 11 (iEdg‘𝐻) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
2610a1i 11 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → 𝐸 ∈ V)
2726resiexd 6974 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → ( I ↾ 𝐸) ∈ V)
28 opiedgfv 26706 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
292, 27, 28sylancr 587 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
3025, 29syl5eq 2873 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (iEdg‘𝐻) = ( I ↾ 𝐸))
3130dmeqd 5773 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = dom ( I ↾ 𝐸))
32 dmresi 5920 . . . . . . . . . 10 dom ( I ↾ 𝐸) = 𝐸
339a1i 11 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → 𝐸 = (Edg‘𝐺))
34 edgval 26748 . . . . . . . . . . 11 (Edg‘𝐺) = ran (iEdg‘𝐺)
3533, 34syl6eq 2877 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
3632, 35syl5eq 2873 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom ( I ↾ 𝐸) = ran (iEdg‘𝐺))
3731, 36eqtrd 2861 . . . . . . . 8 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = ran (iEdg‘𝐺))
3837f1oeq3d 6609 . . . . . . 7 (𝐺 ∈ USHGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺)))
3924, 38mpbird 258 . . . . . 6 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
40 ushgruhgr 26768 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
411, 21uhgrss 26763 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
4240, 41sylan 580 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
43 resiima 5942 . . . . . . . . 9 (((iEdg‘𝐺)‘𝑖) ⊆ 𝑉 → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
4442, 43syl 17 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
45 f1f 6572 . . . . . . . . . . . . 13 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
4622, 45syl 17 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
4746ffund 6515 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
48 fvelrn 6840 . . . . . . . . . . 11 ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
4947, 48sylan 580 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
509, 34eqtri 2849 . . . . . . . . . 10 𝐸 = ran (iEdg‘𝐺)
5149, 50syl6eleqr 2929 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐸)
52 fvresi 6931 . . . . . . . . 9 (((iEdg‘𝐺)‘𝑖) ∈ 𝐸 → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
5351, 52syl 17 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
5410a1i 11 . . . . . . . . . . . 12 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → 𝐸 ∈ V)
5554resiexd 6974 . . . . . . . . . . 11 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) ∈ V)
562, 55, 28sylancr 587 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
5725, 56syl5req 2874 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) = (iEdg‘𝐻))
5857fveq1d 6669 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
5944, 53, 583eqtr2d 2867 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6059ralrimiva 3187 . . . . . 6 (𝐺 ∈ USHGraph → ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6139, 60jca 512 . . . . 5 (𝐺 ∈ USHGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
62 f1oeq1 6601 . . . . . 6 (𝑔 = (iEdg‘𝐺) → (𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
63 fveq1 6666 . . . . . . . . 9 (𝑔 = (iEdg‘𝐺) → (𝑔𝑖) = ((iEdg‘𝐺)‘𝑖))
6463fveq2d 6671 . . . . . . . 8 (𝑔 = (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑔𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6564eqeq2d 2837 . . . . . . 7 (𝑔 = (iEdg‘𝐺) → ((( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
6665ralbidv 3202 . . . . . 6 (𝑔 = (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
6762, 66anbi12d 630 . . . . 5 (𝑔 = (iEdg‘𝐺) → ((𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))) ↔ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))))
6820, 61, 67elabd 3672 . . . 4 (𝐺 ∈ USHGraph → ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))
6919, 68jca 512 . . 3 (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)))))
70 f1oeq1 6601 . . . 4 (𝑓 = ( I ↾ 𝑉) → (𝑓:𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻)))
71 imaeq1 5922 . . . . . . . 8 (𝑓 = ( I ↾ 𝑉) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)))
7271eqeq1d 2828 . . . . . . 7 (𝑓 = ( I ↾ 𝑉) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))
7372ralbidv 3202 . . . . . 6 (𝑓 = ( I ↾ 𝑉) → (∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))
7473anbi2d 628 . . . . 5 (𝑓 = ( I ↾ 𝑉) → ((𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))) ↔ (𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)))))
7574exbidv 1915 . . . 4 (𝑓 = ( I ↾ 𝑉) → (∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))) ↔ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)))))
7670, 75anbi12d 630 . . 3 (𝑓 = ( I ↾ 𝑉) → ((𝑓:𝑉1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)))) ↔ (( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))))
774, 69, 76elabd 3672 . 2 (𝐺 ∈ USHGraph → ∃𝑓(𝑓:𝑉1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)))))
78 opex 5353 . . . 4 𝑉, ( I ↾ 𝐸)⟩ ∈ V
797, 78eqeltri 2914 . . 3 𝐻 ∈ V
80 eqid 2826 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
81 eqid 2826 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
821, 80, 21, 81isomgr 43820 . . 3 ((𝐺 ∈ USHGraph ∧ 𝐻 ∈ V) → (𝐺 IsomGr 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))))
8379, 82mpan2 687 . 2 (𝐺 ∈ USHGraph → (𝐺 IsomGr 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))))
8477, 83mpbird 258 1 (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wral 3143  Vcvv 3500  cdif 3937  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564  cop 4570   class class class wbr 5063   I cid 5458  dom cdm 5554  ran crn 5555  cres 5556  cima 5557  Fun wfun 6346  wf 6348  1-1wf1 6349  1-1-ontowf1o 6351  cfv 6352  Vtxcvtx 26695  iEdgciedg 26696  Edgcedg 26746  UHGraphcuhgr 26755  USHGraphcushgr 26756   IsomGr cisomgr 43816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-1st 7680  df-2nd 7681  df-vtx 26697  df-iedg 26698  df-edg 26747  df-uhgr 26757  df-ushgr 26758  df-isomgr 43818
This theorem is referenced by: (None)
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