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Theorem ushrisomgr 44966
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 6731 . . . . 5 𝑉 ∈ V
32a1i 11 . . . 4 (𝐺 ∈ USHGraph → 𝑉 ∈ V)
43resiexd 7032 . . 3 (𝐺 ∈ USHGraph → ( I ↾ 𝑉) ∈ V)
5 f1oi 6698 . . . . . 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 6720 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
9 ushrisomgr.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
109fvexi 6731 . . . . . . . . . 10 𝐸 ∈ V
11 id 22 . . . . . . . . . . 11 (𝐸 ∈ V → 𝐸 ∈ V)
1211resiexd 7032 . . . . . . . . . 10 (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V)
1310, 12ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐸) ∈ V
142, 13pm3.2i 474 . . . . . . . 8 (𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V)
15 opvtxfv 27095 . . . . . . . 8 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
1614, 15mp1i 13 . . . . . . 7 (𝐺 ∈ USHGraph → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
178, 16syl5eq 2790 . . . . . 6 (𝐺 ∈ USHGraph → (Vtx‘𝐻) = 𝑉)
1817f1oeq3d 6658 . . . . 5 (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto𝑉))
196, 18mpbird 260 . . . 4 (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻))
20 fvexd 6732 . . . . 5 (𝐺 ∈ USHGraph → (iEdg‘𝐺) ∈ V)
21 eqid 2737 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
221, 21ushgrf 27154 . . . . . . . 8 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}))
23 f1f1orn 6672 . . . . . . . 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 6720 . . . . . . . . . . 11 (iEdg‘𝐻) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
2610a1i 11 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → 𝐸 ∈ V)
2726resiexd 7032 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → ( I ↾ 𝐸) ∈ V)
28 opiedgfv 27098 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
292, 27, 28sylancr 590 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
3025, 29syl5eq 2790 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (iEdg‘𝐻) = ( I ↾ 𝐸))
3130dmeqd 5774 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = dom ( I ↾ 𝐸))
32 dmresi 5921 . . . . . . . . . 10 dom ( I ↾ 𝐸) = 𝐸
339a1i 11 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → 𝐸 = (Edg‘𝐺))
34 edgval 27140 . . . . . . . . . . 11 (Edg‘𝐺) = ran (iEdg‘𝐺)
3533, 34eqtrdi 2794 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
3632, 35syl5eq 2790 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom ( I ↾ 𝐸) = ran (iEdg‘𝐺))
3731, 36eqtrd 2777 . . . . . . . 8 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = ran (iEdg‘𝐺))
3837f1oeq3d 6658 . . . . . . 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 260 . . . . . 6 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
40 ushgruhgr 27160 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
411, 21uhgrss 27155 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
4240, 41sylan 583 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
43 resiima 5944 . . . . . . . . 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 6615 . . . . . . . . . . . . 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 6549 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
48 fvelrn 6897 . . . . . . . . . . 11 ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
4947, 48sylan 583 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
509, 34eqtri 2765 . . . . . . . . . 10 𝐸 = ran (iEdg‘𝐺)
5149, 50eleqtrrdi 2849 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐸)
52 fvresi 6988 . . . . . . . . 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 7032 . . . . . . . . . . 11 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) ∈ V)
562, 55, 28sylancr 590 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
5725, 56eqtr2id 2791 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) = (iEdg‘𝐻))
5857fveq1d 6719 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
5944, 53, 583eqtr2d 2783 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6059ralrimiva 3105 . . . . . 6 (𝐺 ∈ USHGraph → ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6139, 60jca 515 . . . . 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 6649 . . . . . 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 6716 . . . . . . . . 9 (𝑔 = (iEdg‘𝐺) → (𝑔𝑖) = ((iEdg‘𝐺)‘𝑖))
6463fveq2d 6721 . . . . . . . 8 (𝑔 = (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑔𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6564eqeq2d 2748 . . . . . . 7 (𝑔 = (iEdg‘𝐺) → ((( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
6665ralbidv 3118 . . . . . 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 634 . . . . 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, 67spcedv 3513 . . . 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 515 . . 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 6649 . . . 4 (𝑓 = ( I ↾ 𝑉) → (𝑓:𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻)))
71 imaeq1 5924 . . . . . . . 8 (𝑓 = ( I ↾ 𝑉) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)))
7271eqeq1d 2739 . . . . . . 7 (𝑓 = ( I ↾ 𝑉) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))
7372ralbidv 3118 . . . . . 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 632 . . . . 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 1929 . . . 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 634 . . 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, 76spcedv 3513 . 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 5348 . . . 4 𝑉, ( I ↾ 𝐸)⟩ ∈ V
797, 78eqeltri 2834 . . 3 𝐻 ∈ V
80 eqid 2737 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
81 eqid 2737 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
821, 80, 21, 81isomgr 44948 . . 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 691 . 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 260 1 (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wral 3061  Vcvv 3408  cdif 3863  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541  cop 4547   class class class wbr 5053   I cid 5454  dom cdm 5551  ran crn 5552  cres 5553  cima 5554  Fun wfun 6374  wf 6376  1-1wf1 6377  1-1-ontowf1o 6379  cfv 6380  Vtxcvtx 27087  iEdgciedg 27088  Edgcedg 27138  UHGraphcuhgr 27147  USHGraphcushgr 27148   IsomGr cisomgr 44944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1st 7761  df-2nd 7762  df-vtx 27089  df-iedg 27090  df-edg 27139  df-uhgr 27149  df-ushgr 27150  df-isomgr 44946
This theorem is referenced by: (None)
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