Theorem incistruhgr 15911

Description: An incidence structure ⟨𝑃, 𝐿, 𝐼⟩ "where 𝑃 is a set whose elements are called points, 𝐿 is a distinct set whose elements are called lines and 𝐼 ⊆ (𝑃 × 𝐿) is the incidence relation" (see Wikipedia "Incidence structure" (24-Oct-2020), https://en.wikipedia.org/wiki/Incidence_structure) implies an undirected hypergraph, if the incidence relation is right-total (to exclude empty edges). The points become the vertices, and the edge function is derived from the incidence relation by mapping each line ("edge") to the set of vertices incident to the line/edge. With 𝑃 = (Base‘𝑆) and by defining two new slots for lines and incidence relations and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incidence structure is an undirected hypergraph. By choosing the incident relation appropriately, other kinds of undirected graphs (pseudographs, multigraphs, simple graphs, etc.) could be defined. (Contributed by AV, 24-Oct-2020.)

Hypotheses

Ref	Expression
incistruhgr.v	⊢ 𝑉 = (Vtx‘𝐺)
incistruhgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
incistruhgr	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝐼,𝑣   𝑒,𝐿,𝑣   𝑃,𝑒,𝑣   𝑒,𝑉,𝑣   𝑒,𝑊

Allowed substitution hints:   𝐸(𝑣)   𝐺(𝑣)   𝑊(𝑣)

Proof of Theorem incistruhgr

Dummy variables 𝑗 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2791	. . . . . . . . 9 ⊢ (𝑉 = 𝑃 → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})
2	1	mpteq2dv 4175	. . . . . . . 8 ⊢ (𝑉 = 𝑃 → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))
3	2	eqeq2d 2241	. . . . . . 7 ⊢ (𝑉 = 𝑃 → (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})))
4		xpeq1 4734	. . . . . . . . 9 ⊢ (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿))
5	4	sseq2d 3254	. . . . . . . 8 ⊢ (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿)))
6	5	3anbi2d 1351	. . . . . . 7 ⊢ (𝑉 = 𝑃 → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ↔ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)))
7	3, 6	anbi12d 473	. . . . . 6 ⊢ (𝑉 = 𝑃 → ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) ↔ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿))))
8		simpl 109	. . . . . . . 8 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
9		dmeq 4926	. . . . . . . . 9 ⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom 𝐸 = dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
10		eqid 2229	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒})
11		eqid 2229	. . . . . . . . . . 11 ⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}
12		incistruhgr.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
13		simpl1 1024	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → 𝐺 ∈ 𝑊)
14		vtxex 15840	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝑊 → (Vtx‘𝐺) ∈ V)
15	13, 14	syl 14	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → (Vtx‘𝐺) ∈ V)
16	12, 15	eqeltrid 2316	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → 𝑉 ∈ V)
17	11, 16	rabexd 4230	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ V)
18	10, 17	dmmptd 5457	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = 𝐿)
19	9, 18	sylan9eq 2282	. . . . . . . 8 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → dom 𝐸 = 𝐿)
20	8, 19	jca 306	. . . . . . 7 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿))
21		simpr 110	. . . . . . 7 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿))
22		eleq2 2293	. . . . . . . . . . 11 ⊢ (𝑠 = {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} → (𝑗 ∈ 𝑠 ↔ 𝑗 ∈ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
23	22	exbidv 1871	. . . . . . . . . 10 ⊢ (𝑠 = {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} → (∃𝑗 𝑗 ∈ 𝑠 ↔ ∃𝑗 𝑗 ∈ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
24		ssrab2 3309	. . . . . . . . . . 11 ⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉
25		elpwg 3657	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ V → ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉))
26	17, 25	syl 14	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉))
27	24, 26	mpbiri 168	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉)
28		eleq2 2293	. . . . . . . . . . . . . 14 ⊢ (ran 𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿))
29	28	3ad2ant3 1044	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿))
30		ssrelrn 4917	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
31	30	ex 115	. . . . . . . . . . . . . 14 ⊢ (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
32	31	3ad2ant2 1043	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
33	29, 32	sylbird 170	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
34	33	imp 124	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
35		rabn0m 3519	. . . . . . . . . . 11 ⊢ (∃𝑗 𝑗 ∈ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
36	34, 35	sylibr 134	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ∃𝑗 𝑗 ∈ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒})
37	23, 27, 36	elrabd 2961	. . . . . . . . 9 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
38	37	fmpttd 5795	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
39		simpl 109	. . . . . . . . 9 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
40		simpr 110	. . . . . . . . 9 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿)
41	39, 40	feq12d 5466	. . . . . . . 8 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → (𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
42	38, 41	imbitrrid 156	. . . . . . 7 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
43	20, 21, 42	sylc 62	. . . . . 6 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
44	7, 43	biimtrrdi 164	. . . . 5 ⊢ (𝑉 = 𝑃 → ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
45	44	expdimp 259	. . . 4 ⊢ ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
46	45	impcom 125	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
47		incistruhgr.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
48	12, 47	isuhgrm 15892	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
49	48	3ad2ant1 1042	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
50	49	adantr 276	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
51	46, 50	mpbird 167	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐺 ∈ UHGraph)
52	51	ex 115	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  {crab 2512  Vcvv 2799   ⊆ wss 3197  𝒫 cpw 3649   class class class wbr 4083   ↦ cmpt 4145   × cxp 4718  dom cdm 4720  ran crn 4721  ⟶wf 5317  ‘cfv 5321  Vtxcvtx 15834  iEdgciedg 15835  UHGraphcuhgr 15888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fo 5327  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-sub 8335  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-dec 9595  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-uhgrm 15890

This theorem is referenced by: (None)