Theorem frgrncvvdeqlem9 29251

Description: Lemma 9 for frgrncvvdeq 29253. 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 29246	. 2 ⊢ (𝜑 → 𝐴:𝐷⟶𝑁)
12	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐺 ∈ FriendGraph )
13	4	eleq2i 2829	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑁 ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑌))
14	1	nbgrisvtx 28289	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑉)
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑉))
16	13, 15	biimtrid 241	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑁 → 𝑛 ∈ 𝑉))
17	16	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑛 ∈ 𝑉)
18	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑋 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem1 29243	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
20		df-nel 3050	. . . . . . . . . . 11 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
21		nelelne 3043	. . . . . . . . . . 11 ⊢ (¬ 𝑋 ∈ 𝑁 → (𝑛 ∈ 𝑁 → 𝑛 ≠ 𝑋))
22	20, 21	sylbi 216	. . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 → (𝑛 ∈ 𝑁 → 𝑛 ≠ 𝑋))
23	19, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ 𝑁 → 𝑛 ≠ 𝑋))
24	23	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑛 ≠ 𝑋)
25	17, 18, 24	3jca 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋))
26	12, 25	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋)))
27	1, 2	frcond2 29211	. . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → ((𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋) → ∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
28	27	imp 407	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑛 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑛 ≠ 𝑋)) → ∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
29		reurex 3357	. . . . . . 7 ⊢ (∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
30		df-rex 3074	. . . . . . 7 ⊢ (∃𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
31	29, 30	sylib 217	. . . . . 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 29205	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
34	2	nbusgreledg 28301	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
35	34	bicomd 222	. . . . . . . . . . . . 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 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
38	3	eleq2i 2829	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
39	37, 38	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ 𝐷)
40	39	ad2ant2rl 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑚 ∈ 𝐷)
41	2	nbusgreledg 28301	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ↔ {𝑛, 𝑚} ∈ 𝐸))
42	41	biimpar 478	. . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
47	46	imp 407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
48		elin 3926	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛 ∈ 𝑁))
49		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝜑)
50	49, 39	jca 512	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝜑 ∧ 𝑚 ∈ 𝐷))
51		preq1 4694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
52	51	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
53	52	riotabidv 7315	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑚 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
54	53	cbvmptv 5218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
55	10, 54	eqtri 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 = (𝑚 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 55	frgrncvvdeqlem5 29247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐷) → {(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
57		eleq2 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴‘𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴‘𝑚)}))
58	57	eqcoms 2744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴‘𝑚)}))
59		elsni 4603	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ {(𝐴‘𝑚)} → 𝑛 = (𝐴‘𝑚))
60	58, 59	syl6bi 252	. . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛 ∈ 𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑 → 𝑛 = (𝐴‘𝑚))))
65	64	ex 413	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴‘𝑚))))
68	67	adantld 491	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴‘𝑚))))
69	68	imp 407	. . . . . . . . . 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 512	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚)))
72	71	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚))))
73	72	adantld 491	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ((𝑚 ∈ 𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚))))
74	73	eximdv 1920	. . . . 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 3074	. . . 4 ⊢ (∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚) ↔ ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = (𝐴‘𝑚)))
77	75, 76	sylibr 233	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚))
78	77	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚))
79		dffo3 7052	. 2 ⊢ (𝐴:𝐷–onto→𝑁 ↔ (𝐴:𝐷⟶𝑁 ∧ ∀𝑛 ∈ 𝑁 ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚)))
80	11, 78, 79	sylanbrc 583	1 ⊢ (𝜑 → 𝐴:𝐷–onto→𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  ∃wrex 3073  ∃!wreu 3351   ∩ cin 3909  {csn 4586  {cpr 4588   ↦ cmpt 5188  ⟶wf 6492  –onto→wfo 6494  ‘cfv 6496  ℩crio 7312  (class class class)co 7357  Vtxcvtx 27947  Edgcedg 27998  USGraphcusgr 28100   NeighbVtx cnbgr 28280   FriendGraph cfrgr 29202

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-upgr 28033  df-umgr 28034  df-usgr 28102  df-nbgr 28281  df-frgr 29203

This theorem is referenced by:  frgrncvvdeqlem10  29252