Theorem ushggricedg 48115

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 → 𝐺 ≃𝑔𝑟 𝐻)

Proof of Theorem ushggricedg

Dummy variables 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ushggricedg.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6846	. . . . 5 ⊢ 𝑉 ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝑉 ∈ V)
4	3	resiexd 7160	. . 3 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝑉) ∈ V)
5		f1oi 6810	. . . . . 6 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
6	5	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉–1-1-onto→𝑉)
7		ushggricedg.s	. . . . . . . 8 ⊢ 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩
8	7	fveq2i 6835	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
9		ushggricedg.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
10	9	fvexi 6846	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
11		resiexg 7852	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ 𝐸) ∈ V
13	2, 12	pm3.2i 470	. . . . . . . 8 ⊢ (𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V)
14		opvtxfv 29026	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
15	13, 14	mp1i 13	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
16	8, 15	eqtrid 2781	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (Vtx‘𝐻) = 𝑉)
17	16	f1oeq3d 6769	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉))
18	6, 17	mpbird 257	. . . 4 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻))
19		fvexd 6847	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) ∈ V)
20		eqid 2734	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
21	1, 20	ushgrf 29085	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}))
22		f1f1orn 6783	. . . . . . . 8 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
24	7	fveq2i 6835	. . . . . . . . . . 11 ⊢ (iEdg‘𝐻) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
25	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → 𝐸 ∈ V)
26	25	resiexd 7160	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝐸) ∈ V)
27		opiedgfv 29029	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
28	2, 26, 27	sylancr 587	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
29	24, 28	eqtrid 2781	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐻) = ( I ↾ 𝐸))
30	29	dmeqd 5852	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = dom ( I ↾ 𝐸))
31		dmresi 6009	. . . . . . . . . 10 ⊢ dom ( I ↾ 𝐸) = 𝐸
32	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → 𝐸 = (Edg‘𝐺))
33		edgval 29071	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
34	32, 33	eqtrdi 2785	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
35	31, 34	eqtrid 2781	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → dom ( I ↾ 𝐸) = ran (iEdg‘𝐺))
36	30, 35	eqtrd 2769	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = ran (iEdg‘𝐺))
37	36	f1oeq3d 6769	. . . . . . 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)))
38	23, 37	mpbird 257	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
39		ushgruhgr 29091	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
40	1, 20	uhgrss 29086	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
41	39, 40	sylan 580	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
42		resiima 6033	. . . . . . . . 9 ⊢ (((iEdg‘𝐺)‘𝑖) ⊆ 𝑉 → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
43	41, 42	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
44		f1f 6728	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
45	21, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
46	45	ffund 6664	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
47		fvelrn 7019	. . . . . . . . . . 11 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
48	46, 47	sylan 580	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
49	9, 33	eqtri 2757	. . . . . . . . . 10 ⊢ 𝐸 = ran (iEdg‘𝐺)
50	48, 49	eleqtrrdi 2845	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐸)
51		fvresi 7117	. . . . . . . . 9 ⊢ (((iEdg‘𝐺)‘𝑖) ∈ 𝐸 → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
52	50, 51	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
53	10	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → 𝐸 ∈ V)
54	53	resiexd 7160	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) ∈ V)
55	2, 54, 27	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
56	24, 55	eqtr2id 2782	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) = (iEdg‘𝐻))
57	56	fveq1d 6834	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
58	43, 52, 57	3eqtr2d 2775	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
59	58	ralrimiva 3126	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
60	38, 59	jca 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 6760	. . . . . 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 6831	. . . . . . . . 9 ⊢ (𝑔 = (iEdg‘𝐺) → (𝑔‘𝑖) = ((iEdg‘𝐺)‘𝑖))
63	62	fveq2d 6836	. . . . . . . 8 ⊢ (𝑔 = (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑔‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
64	63	eqeq2d 2745	. . . . . . 7 ⊢ (𝑔 = (iEdg‘𝐺) → ((( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
65	64	ralbidv 3157	. . . . . 6 ⊢ (𝑔 = (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
66	61, 65	anbi12d 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)‘(𝑔‘𝑖))) ↔ ((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)))))
67	19, 60, 66	spcedv 3550	. . . 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)‘(𝑔‘𝑖))))
68	18, 67	jca 511	. . 3 ⊢ (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉–1-1-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)‘(𝑔‘𝑖)))))
69		f1oeq1 6760	. . . 4 ⊢ (𝑓 = ( I ↾ 𝑉) → (𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻)))
70		imaeq1 6012	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ 𝑉) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)))
71	70	eqeq1d 2736	. . . . . . 7 ⊢ (𝑓 = ( I ↾ 𝑉) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))
72	71	ralbidv 3157	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝑉) → (∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))
73	72	anbi2d 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)‘(𝑔‘𝑖))) ↔ (𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
74	73	exbidv 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)‘(𝑔‘𝑖))) ↔ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
75	69, 74	anbi12d 632	. . 3 ⊢ (𝑓 = ( I ↾ 𝑉) → ((𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))) ↔ (( I ↾ 𝑉):𝑉–1-1-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)‘(𝑔‘𝑖))))))
76	4, 68, 75	spcedv 3550	. 2 ⊢ (𝐺 ∈ USHGraph → ∃𝑓(𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
77		opex 5410	. . . 4 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
78	7, 77	eqeltri 2830	. . 3 ⊢ 𝐻 ∈ V
79		eqid 2734	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
80		eqid 2734	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
81	1, 79, 20, 80	dfgric2 48103	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝐻 ∈ V) → (𝐺 ≃𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))))
82	78, 81	mpan2 691	. 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 ≃𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))))
83	76, 82	mpbird 257	1 ⊢ (𝐺 ∈ USHGraph → 𝐺 ≃𝑔𝑟 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  {csn 4578  ⟨cop 4584   class class class wbr 5096   I cid 5516  dom cdm 5622  ran crn 5623   ↾ cres 5624   “ cima 5625  Fun wfun 6484  ⟶wf 6486  –1-1→wf1 6487  –1-1-onto→wf1o 6489  ‘cfv 6490  Vtxcvtx 29018  iEdgciedg 29019  Edgcedg 29069  UHGraphcuhgr 29078  USHGraphcushgr 29079   ≃_𝑔𝑟 cgric 48064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-1o 8395  df-map 8763  df-vtx 29020  df-iedg 29021  df-edg 29070  df-uhgr 29080  df-ushgr 29081  df-grim 48066  df-gric 48069

This theorem is referenced by: (None)