Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ushrisomgr Structured version   Visualization version   GIF version

Theorem ushrisomgr 44402
 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 6660 . . . . 5 𝑉 ∈ V
32a1i 11 . . . 4 (𝐺 ∈ USHGraph → 𝑉 ∈ V)
43resiexd 6957 . . 3 (𝐺 ∈ USHGraph → ( I ↾ 𝑉) ∈ V)
5 f1oi 6628 . . . . . 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 6649 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
9 ushrisomgr.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
109fvexi 6660 . . . . . . . . . 10 𝐸 ∈ V
11 id 22 . . . . . . . . . . 11 (𝐸 ∈ V → 𝐸 ∈ V)
1211resiexd 6957 . . . . . . . . . 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 26807 . . . . . . . 8 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
1614, 15mp1i 13 . . . . . . 7 (𝐺 ∈ USHGraph → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
178, 16syl5eq 2845 . . . . . 6 (𝐺 ∈ USHGraph → (Vtx‘𝐻) = 𝑉)
1817f1oeq3d 6588 . . . . 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 6661 . . . . 5 (𝐺 ∈ USHGraph → (iEdg‘𝐺) ∈ V)
21 eqid 2798 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
221, 21ushgrf 26866 . . . . . . . 8 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}))
23 f1f1orn 6602 . . . . . . . 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 6649 . . . . . . . . . . 11 (iEdg‘𝐻) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
2610a1i 11 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → 𝐸 ∈ V)
2726resiexd 6957 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → ( I ↾ 𝐸) ∈ V)
28 opiedgfv 26810 . . . . . . . . . . . 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 2845 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (iEdg‘𝐻) = ( I ↾ 𝐸))
3130dmeqd 5739 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = dom ( I ↾ 𝐸))
32 dmresi 5889 . . . . . . . . . 10 dom ( I ↾ 𝐸) = 𝐸
339a1i 11 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → 𝐸 = (Edg‘𝐺))
34 edgval 26852 . . . . . . . . . . 11 (Edg‘𝐺) = ran (iEdg‘𝐺)
3533, 34eqtrdi 2849 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
3632, 35syl5eq 2845 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom ( I ↾ 𝐸) = ran (iEdg‘𝐺))
3731, 36eqtrd 2833 . . . . . . . 8 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = ran (iEdg‘𝐺))
3837f1oeq3d 6588 . . . . . . 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 26872 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
411, 21uhgrss 26867 . . . . . . . . . 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 5912 . . . . . . . . 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 6550 . . . . . . . . . . . . 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 6492 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
48 fvelrn 6822 . . . . . . . . . . 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 2821 . . . . . . . . . 10 𝐸 = ran (iEdg‘𝐺)
5149, 50eleqtrrdi 2901 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐸)
52 fvresi 6913 . . . . . . . . 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 6957 . . . . . . . . . . 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, 56syl5req 2846 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) = (iEdg‘𝐻))
5857fveq1d 6648 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
5944, 53, 583eqtr2d 2839 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6059ralrimiva 3149 . . . . . 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 6580 . . . . . 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 6645 . . . . . . . . 9 (𝑔 = (iEdg‘𝐺) → (𝑔𝑖) = ((iEdg‘𝐺)‘𝑖))
6463fveq2d 6650 . . . . . . . 8 (𝑔 = (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑔𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6564eqeq2d 2809 . . . . . . 7 (𝑔 = (iEdg‘𝐺) → ((( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
6665ralbidv 3162 . . . . . 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 633 . . . . 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 3547 . . . 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 6580 . . . 4 (𝑓 = ( I ↾ 𝑉) → (𝑓:𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻)))
71 imaeq1 5892 . . . . . . . 8 (𝑓 = ( I ↾ 𝑉) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)))
7271eqeq1d 2800 . . . . . . 7 (𝑓 = ( I ↾ 𝑉) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))
7372ralbidv 3162 . . . . . 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 631 . . . . 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 1922 . . . 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 633 . . 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 3547 . 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 5322 . . . 4 𝑉, ( I ↾ 𝐸)⟩ ∈ V
797, 78eqeltri 2886 . . 3 𝐻 ∈ V
80 eqid 2798 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
81 eqid 2798 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
821, 80, 21, 81isomgr 44384 . . 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 690 . 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 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ⟨cop 4531   class class class wbr 5031   I cid 5425  dom cdm 5520  ran crn 5521   ↾ cres 5522   “ cima 5523  Fun wfun 6319  ⟶wf 6321  –1-1→wf1 6322  –1-1-onto→wf1o 6324  ‘cfv 6325  Vtxcvtx 26799  iEdgciedg 26800  Edgcedg 26850  UHGraphcuhgr 26859  USHGraphcushgr 26860   IsomGr cisomgr 44380 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-1st 7674  df-2nd 7675  df-vtx 26801  df-iedg 26802  df-edg 26851  df-uhgr 26861  df-ushgr 26862  df-isomgr 44382 This theorem is referenced by: (None)
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