Theorem frgrncvvdeqlem9 30363

Description: Lemma 9 for frgrncvvdeq 30365. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e	⊢ 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny	⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem9	⊢ (𝜑 → 𝐴:𝐷–onto→𝑁)

Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥   𝑦,𝐷   𝑥,𝐸

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem9

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrncvvdeq.v1	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrncvvdeq.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3		frgrncvvdeq.nx	. . 3 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4		frgrncvvdeq.ny	. . 3 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
5		frgrncvvdeq.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		frgrncvvdeq.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
7		frgrncvvdeq.ne	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
8		frgrncvvdeq.xy	. . 3 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
9		frgrncvvdeq.f	. . 3 ⊢ (𝜑 → 𝐺 ∈ FriendGraph )
10		frgrncvvdeq.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem4 30358	. 2 ⊢ (𝜑 → 𝐴:𝐷⟶𝑁)
12	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐺 ∈ FriendGraph )
13	4	eleq2i 2827	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑁 ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑌))
14	1	nbgrisvtx 29395	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑉)
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑉))
16	13, 15	biimtrid 242	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑁 → 𝑛 ∈ 𝑉))
17	16	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑛 ∈ 𝑉)
18	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑋 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem1 30355	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
20		df-nel 3036	. . . . . . . . . . 11 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
21		nelelne 3030	. . . . . . . . . . 11 ⊢ (¬ 𝑋 ∈ 𝑁 → (𝑛 ∈ 𝑁 → 𝑛 ≠ 𝑋))
22	20, 21	sylbi 217	. . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 → (𝑛 ∈ 𝑁 → 𝑛 ≠ 𝑋))
23	19, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑁 → 𝑛 ≠ 𝑋))
24	23	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑛 ≠ 𝑋)
25	17, 18, 24	3jca 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋))
26	12, 25	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋)))
27	1, 2	frcond2 30323	. . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋) → ∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
28	27	imp 406	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋)) → ∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
29		reurex 3353	. . . . . . 7 ⊢ (∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
30		df-rex 3060	. . . . . . 7 ⊢ (∃𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
31	29, 30	sylib 218	. . . . . 6 ⊢ (∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚(𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
32	26, 28, 31	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∃𝑚(𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
33		frgrusgr 30317	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
34	2	nbusgreledg 29407	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
35	34	bicomd 223	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → ({𝑚, 𝑋} ∈ 𝐸 ↔ 𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
36	9, 33, 35	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ({𝑚, 𝑋} ∈ 𝐸 ↔ 𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
37	36	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
38	3	eleq2i 2827	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
39	37, 38	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ 𝐷)
40	39	ad2ant2rl 750	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑚 ∈ 𝐷)
41	2	nbusgreledg 29407	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ↔ {𝑛, 𝑚} ∈ 𝐸))
42	41	biimpar 477	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ {𝑛, 𝑚} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
43	42	a1d 25	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ {𝑛, 𝑚} ∈ 𝐸) → ({𝑚, 𝑋} ∈ 𝐸 → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
44	43	expimpd 453	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
45	9, 33, 44	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
46	45	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
47	46	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
48		elin 3916	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛 ∈ 𝑁))
49		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝜑)
50	49, 39	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝜑 ∧ 𝑚 ∈ 𝐷))
51		preq1 4689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
52	51	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
53	52	riotabidv 7317	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑚 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
54	53	cbvmptv 5201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
55	10, 54	eqtri 2758	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 = (𝑚 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 55	frgrncvvdeqlem5 30359	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐷) → {(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
57		eleq2 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴‘𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴‘𝑚)}))
58	57	eqcoms 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴‘𝑚)}))
59		elsni 4596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ {(𝐴‘𝑚)} → 𝑛 = (𝐴‘𝑚))
60	58, 59	biimtrdi 253	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴‘𝑚)))
61	50, 56, 60	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴‘𝑚)))
62	61	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑚, 𝑋} ∈ 𝐸 → (𝜑 → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴‘𝑚))))
63	62	com3r 87	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑 → 𝑛 = (𝐴‘𝑚))))
64	48, 63	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛 ∈ 𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑 → 𝑛 = (𝐴‘𝑚))))
65	64	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → (𝑛 ∈ 𝑁 → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑 → 𝑛 = (𝐴‘𝑚)))))
66	65	com14 96	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ 𝑁 → ({𝑚, 𝑋} ∈ 𝐸 → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴‘𝑚)))))
67	66	imp 406	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴‘𝑚))))
68	67	adantld 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴‘𝑚))))
69	68	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴‘𝑚)))
70	47, 69	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 = (𝐴‘𝑚))
71	40, 70	jca 511	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚)))
72	71	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚))))
73	72	adantld 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ((𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚))))
74	73	eximdv 1919	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (∃𝑚(𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚))))
75	32, 74	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚)))
76		df-rex 3060	. . . 4 ⊢ (∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚) ↔ ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚)))
77	75, 76	sylibr 234	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚))
78	77	ralrimiva 3127	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚))
79		dffo3 7047	. 2 ⊢ (𝐴:𝐷–onto→𝑁 ↔ (𝐴:𝐷⟶𝑁 ∧ ∀𝑛 ∈ 𝑁 ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚)))
80	11, 78, 79	sylanbrc 584	1 ⊢ (𝜑 → 𝐴:𝐷–onto→𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2931   ∉ wnel 3035  ∀wral 3050  ∃wrex 3059  ∃!wreu 3347   ∩ cin 3899  {csn 4579  {cpr 4581   ↦ cmpt 5178  ⟶wf 6487  –onto→wfo 6489  ‘cfv 6491  ℩crio 7314  (class class class)co 7358  Vtxcvtx 29050  Edgcedg 29101  USGraphcusgr 29203   NeighbVtx cnbgr 29386   FriendGraph cfrgr 30314

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-hash 14256  df-edg 29102  df-upgr 29136  df-umgr 29137  df-usgr 29205  df-nbgr 29387  df-frgr 30315

This theorem is referenced by:  frgrncvvdeqlem10  30364