Theorem upgredg 25811

Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)

Hypotheses

Ref	Expression
upgredg.v	⊢ 𝑉 = (Vtx‘𝐺)
upgredg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
upgredg	⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})

Distinct variable groups:   𝐶,𝑎,𝑏   𝐺,𝑎,𝑏   𝑉,𝑎,𝑏

Allowed substitution hints:   𝐸(𝑎,𝑏)

Proof of Theorem upgredg

Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
2		edgaval 25794	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	syl5eq 2656	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4	3	eleq2d 2673	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ran (iEdg‘𝐺)))
5		upgredg.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6		eqid 2610	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7	5, 6	upgrf 25753	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
8		frn 5966	. . . . . . 7 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
9	7, 8	syl 17	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
10	9	sseld 3567	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ ran (iEdg‘𝐺) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
11	4, 10	sylbid 229	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ 𝐸 → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
12	11	imp 444	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
13		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝐶 → (#‘𝑥) = (#‘𝐶))
14	13	breq1d 4593	. . . . 5 ⊢ (𝑥 = 𝐶 → ((#‘𝑥) ≤ 2 ↔ (#‘𝐶) ≤ 2))
15	14	elrab 3331	. . . 4 ⊢ (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2))
16		2nn0 11186	. . . . . . 7 ⊢ 2 ∈ ℕ0
17		hashbnd 12985	. . . . . . 7 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ∈ ℕ0 ∧ (#‘𝐶) ≤ 2) → 𝐶 ∈ Fin)
18	16, 17	mp3an2 1404	. . . . . 6 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → 𝐶 ∈ Fin)
19		hashcl 13009	. . . . . 6 ⊢ (𝐶 ∈ Fin → (#‘𝐶) ∈ ℕ0)
20	18, 19	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → (#‘𝐶) ∈ ℕ0)
21		nn0re 11178	. . . . . . . . 9 ⊢ ((#‘𝐶) ∈ ℕ0 → (#‘𝐶) ∈ ℝ)
22		2re 10967	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
23	22	a1i 11	. . . . . . . . 9 ⊢ ((#‘𝐶) ∈ ℕ0 → 2 ∈ ℝ)
24	21, 23	leloed 10059	. . . . . . . 8 ⊢ ((#‘𝐶) ∈ ℕ0 → ((#‘𝐶) ≤ 2 ↔ ((#‘𝐶) < 2 ∨ (#‘𝐶) = 2)))
25	24	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) ≤ 2 ↔ ((#‘𝐶) < 2 ∨ (#‘𝐶) = 2)))
26		nn0lt2 11317	. . . . . . . . . . 11 ⊢ (((#‘𝐶) ∈ ℕ0 ∧ (#‘𝐶) < 2) → ((#‘𝐶) = 0 ∨ (#‘𝐶) = 1))
27	26	ex 449	. . . . . . . . . 10 ⊢ ((#‘𝐶) ∈ ℕ0 → ((#‘𝐶) < 2 → ((#‘𝐶) = 0 ∨ (#‘𝐶) = 1)))
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) < 2 → ((#‘𝐶) = 0 ∨ (#‘𝐶) = 1)))
29		eldifsn 4260	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ↔ (𝐶 ∈ 𝒫 𝑉 ∧ 𝐶 ≠ ∅))
30		hasheq0 13015	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝒫 𝑉 → ((#‘𝐶) = 0 ↔ 𝐶 = ∅))
31	30	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ 𝒫 𝑉 ∧ 𝐶 ≠ ∅) → ((#‘𝐶) = 0 ↔ 𝐶 = ∅))
32		eqneqall 2793	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = ∅ → (𝐶 ≠ ∅ → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
33	32	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝐶 ≠ ∅ → (𝐶 = ∅ → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
34	33	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ 𝒫 𝑉 ∧ 𝐶 ≠ ∅) → (𝐶 = ∅ → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
35	31, 34	sylbid 229	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝒫 𝑉 ∧ 𝐶 ≠ ∅) → ((#‘𝐶) = 0 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
36	29, 35	sylbi 206	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 0 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
37	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) = 0 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
38		hash1snb 13068	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 1 ↔ ∃𝑐 𝐶 = {𝑐}))
39		eleq1 2676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 = {𝑐} → (𝐶 ∈ 𝒫 𝑉 ↔ {𝑐} ∈ 𝒫 𝑉))
40		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑐 ∈ V
41	40	snelpw 4840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ 𝑉 ↔ {𝑐} ∈ 𝒫 𝑉)
42	41	biimpri 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑐} ∈ 𝒫 𝑉 → 𝑐 ∈ 𝑉)
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 = {𝑐} → ({𝑐} ∈ 𝒫 𝑉 → 𝑐 ∈ 𝑉))
44	39, 43	sylbid 229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 = {𝑐} → (𝐶 ∈ 𝒫 𝑉 → 𝑐 ∈ 𝑉))
45	44	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ 𝒫 𝑉 → (𝐶 = {𝑐} → 𝑐 ∈ 𝑉))
46	45	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ 𝒫 𝑉 ∧ 𝐶 ≠ ∅) → (𝐶 = {𝑐} → 𝑐 ∈ 𝑉))
47	29, 46	sylbi 206	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → (𝐶 = {𝑐} → 𝑐 ∈ 𝑉))
48	47	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → 𝑐 ∈ 𝑉)
49		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = {𝑐} → 𝐶 = {𝑐})
50		dfsn2 4138	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑐} = {𝑐, 𝑐}
51	49, 50	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 = {𝑐} → 𝐶 = {𝑐, 𝑐})
52	51	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → 𝐶 = {𝑐, 𝑐})
53		preq1 4212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑐 → {𝑎, 𝑏} = {𝑐, 𝑏})
54	53	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑐 → (𝐶 = {𝑎, 𝑏} ↔ 𝐶 = {𝑐, 𝑏}))
55		preq2 4213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑐 → {𝑐, 𝑏} = {𝑐, 𝑐})
56	55	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑐 → (𝐶 = {𝑐, 𝑏} ↔ 𝐶 = {𝑐, 𝑐}))
57	54, 56	rspc2ev 3295	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐶 = {𝑐, 𝑐}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
58	48, 48, 52, 57	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
59		r2ex 3043	. . . . . . . . . . . . . . 15 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
60	58, 59	sylib 207	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
61	60	ex 449	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → (𝐶 = {𝑐} → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
62	61	exlimdv 1848	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → (∃𝑐 𝐶 = {𝑐} → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
63	38, 62	sylbid 229	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 1 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
64	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) = 1 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
65	37, 64	jaod 394	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → (((#‘𝐶) = 0 ∨ (#‘𝐶) = 1) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
66	28, 65	syld 46	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) < 2 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
67		hash2sspr 13124	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ 𝒫 𝑉 ∧ (#‘𝐶) = 2) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
68	67	ex 449	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝒫 𝑉 → ((#‘𝐶) = 2 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}))
69	68	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝒫 𝑉 ∧ 𝐶 ≠ ∅) → ((#‘𝐶) = 2 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}))
70	29, 69	sylbi 206	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 2 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}))
71	70, 59	syl6ib 240	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 2 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
72	71	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) = 2 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
73	66, 72	jaod 394	. . . . . . 7 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → (((#‘𝐶) < 2 ∨ (#‘𝐶) = 2) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
74	25, 73	sylbid 229	. . . . . 6 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) ≤ 2 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
75	74	impancom 455	. . . . 5 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → ((#‘𝐶) ∈ ℕ0 → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
76	20, 75	mpd 15	. . . 4 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
77	15, 76	sylbi 206	. . 3 ⊢ (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
78	12, 77	syl 17	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
79	78, 59	sylibr 223	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   < clt 9953   ≤ cle 9954  2c2 10947  ℕ₀cn0 11169  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  Edgcedga 25792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-edga 25793

This theorem is referenced by:  upgredg2vtx  25814  upgredgpr  25815