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Theorem ushggricedg 47896
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
ushggricedg.v 𝑉 = (Vtx‘𝐺)
ushggricedg.e 𝐸 = (Edg‘𝐺)
ushggricedg.s 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩
Assertion
Ref Expression
ushggricedg (𝐺 ∈ USHGraph → 𝐺𝑔𝑟 𝐻)

Proof of Theorem ushggricedg
Dummy variables 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ushggricedg.v . . . . . 6 𝑉 = (Vtx‘𝐺)
21fvexi 6920 . . . . 5 𝑉 ∈ V
32a1i 11 . . . 4 (𝐺 ∈ USHGraph → 𝑉 ∈ V)
43resiexd 7236 . . 3 (𝐺 ∈ USHGraph → ( I ↾ 𝑉) ∈ V)
5 f1oi 6886 . . . . . 6 ( I ↾ 𝑉):𝑉1-1-onto𝑉
65a1i 11 . . . . 5 (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉1-1-onto𝑉)
7 ushggricedg.s . . . . . . . 8 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩
87fveq2i 6909 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
9 ushggricedg.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
109fvexi 6920 . . . . . . . . . 10 𝐸 ∈ V
11 resiexg 7934 . . . . . . . . . 10 (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V)
1210, 11ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐸) ∈ V
132, 12pm3.2i 470 . . . . . . . 8 (𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V)
14 opvtxfv 29021 . . . . . . . 8 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
1513, 14mp1i 13 . . . . . . 7 (𝐺 ∈ USHGraph → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
168, 15eqtrid 2789 . . . . . 6 (𝐺 ∈ USHGraph → (Vtx‘𝐻) = 𝑉)
1716f1oeq3d 6845 . . . . 5 (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto𝑉))
186, 17mpbird 257 . . . 4 (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻))
19 fvexd 6921 . . . . 5 (𝐺 ∈ USHGraph → (iEdg‘𝐺) ∈ V)
20 eqid 2737 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
211, 20ushgrf 29080 . . . . . . . 8 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}))
22 f1f1orn 6859 . . . . . . . 8 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
2321, 22syl 17 . . . . . . 7 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
247fveq2i 6909 . . . . . . . . . . 11 (iEdg‘𝐻) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
2510a1i 11 . . . . . . . . . . . . 13 (𝐺 ∈ USHGraph → 𝐸 ∈ V)
2625resiexd 7236 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → ( I ↾ 𝐸) ∈ V)
27 opiedgfv 29024 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
282, 26, 27sylancr 587 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
2924, 28eqtrid 2789 . . . . . . . . . 10 (𝐺 ∈ USHGraph → (iEdg‘𝐻) = ( I ↾ 𝐸))
3029dmeqd 5916 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = dom ( I ↾ 𝐸))
31 dmresi 6070 . . . . . . . . . 10 dom ( I ↾ 𝐸) = 𝐸
329a1i 11 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → 𝐸 = (Edg‘𝐺))
33 edgval 29066 . . . . . . . . . . 11 (Edg‘𝐺) = ran (iEdg‘𝐺)
3432, 33eqtrdi 2793 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
3531, 34eqtrid 2789 . . . . . . . . 9 (𝐺 ∈ USHGraph → dom ( I ↾ 𝐸) = ran (iEdg‘𝐺))
3630, 35eqtrd 2777 . . . . . . . 8 (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = ran (iEdg‘𝐺))
3736f1oeq3d 6845 . . . . . . 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)))
3823, 37mpbird 257 . . . . . 6 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
39 ushgruhgr 29086 . . . . . . . . . 10 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
401, 20uhgrss 29081 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
4139, 40sylan 580 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
42 resiima 6094 . . . . . . . . 9 (((iEdg‘𝐺)‘𝑖) ⊆ 𝑉 → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
4341, 42syl 17 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
44 f1f 6804 . . . . . . . . . . . . 13 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
4521, 44syl 17 . . . . . . . . . . . 12 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
4645ffund 6740 . . . . . . . . . . 11 (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
47 fvelrn 7096 . . . . . . . . . . 11 ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
4846, 47sylan 580 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
499, 33eqtri 2765 . . . . . . . . . 10 𝐸 = ran (iEdg‘𝐺)
5048, 49eleqtrrdi 2852 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐸)
51 fvresi 7193 . . . . . . . . 9 (((iEdg‘𝐺)‘𝑖) ∈ 𝐸 → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
5250, 51syl 17 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
5310a1i 11 . . . . . . . . . . . 12 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → 𝐸 ∈ V)
5453resiexd 7236 . . . . . . . . . . 11 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) ∈ V)
552, 54, 27sylancr 587 . . . . . . . . . 10 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
5624, 55eqtr2id 2790 . . . . . . . . 9 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) = (iEdg‘𝐻))
5756fveq1d 6908 . . . . . . . 8 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
5843, 52, 573eqtr2d 2783 . . . . . . 7 ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
5958ralrimiva 3146 . . . . . 6 (𝐺 ∈ USHGraph → ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6038, 59jca 511 . . . . 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))))
61 f1oeq1 6836 . . . . . 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)))
62 fveq1 6905 . . . . . . . . 9 (𝑔 = (iEdg‘𝐺) → (𝑔𝑖) = ((iEdg‘𝐺)‘𝑖))
6362fveq2d 6910 . . . . . . . 8 (𝑔 = (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑔𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
6463eqeq2d 2748 . . . . . . 7 (𝑔 = (iEdg‘𝐺) → ((( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
6564ralbidv 3178 . . . . . 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))))
6661, 65anbi12d 632 . . . . 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)))))
6719, 60, 66spcedv 3598 . . . 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))))
6818, 67jca 511 . . 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)))))
69 f1oeq1 6836 . . . 4 (𝑓 = ( I ↾ 𝑉) → (𝑓:𝑉1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉1-1-onto→(Vtx‘𝐻)))
70 imaeq1 6073 . . . . . . . 8 (𝑓 = ( I ↾ 𝑉) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)))
7170eqeq1d 2739 . . . . . . 7 (𝑓 = ( I ↾ 𝑉) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔𝑖))))
7271ralbidv 3178 . . . . . 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))))
7372anbi2d 630 . . . . 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)))))
7473exbidv 1921 . . . 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)))))
7569, 74anbi12d 632 . . 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))))))
764, 68, 75spcedv 3598 . 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)))))
77 opex 5469 . . . 4 𝑉, ( I ↾ 𝐸)⟩ ∈ V
787, 77eqeltri 2837 . . 3 𝐻 ∈ V
79 eqid 2737 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
80 eqid 2737 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
811, 79, 20, 80dfgric2 47884 . . 3 ((𝐺 ∈ USHGraph ∧ 𝐻 ∈ V) → (𝐺𝑔𝑟 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))))))
8278, 81mpan2 691 . 2 (𝐺 ∈ USHGraph → (𝐺𝑔𝑟 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))))))
8376, 82mpbird 257 1 (𝐺 ∈ USHGraph → 𝐺𝑔𝑟 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  cop 4632   class class class wbr 5143   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073  USHGraphcushgr 29074  𝑔𝑟 cgric 47862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-ushgr 29076  df-grim 47864  df-gric 47867
This theorem is referenced by: (None)
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