Theorem frgrncvvdeqlem9 28572

Description: Lemma 9 for frgrncvvdeq 28574. 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 28567	. 2 ⊢ (𝜑 → 𝐴:𝐷⟶𝑁)
12	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐺 ∈ FriendGraph )
13	4	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑁 ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑌))
14	1	nbgrisvtx 27611	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑉)
15	14	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑉))
16	13, 15	syl5bi 241	. . . . . . . . 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 28564	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
20		df-nel 3049	. . . . . . . . . . 11 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
21		nelelne 3042	. . . . . . . . . . 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 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑛 ≠ 𝑋)
25	17, 18, 24	3jca 1126	. . . . . . 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 28532	. . . . . . 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 3352	. . . . . . 7 ⊢ (∃!𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚 ∈ 𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
30		df-rex 3069	. . . . . . 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 28526	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
34	2	nbusgreledg 27623	. . . . . . . . . . . . . 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 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
38	3	eleq2i 2830	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
39	37, 38	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ 𝐷)
40	39	ad2ant2rl 745	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑚 ∈ 𝐷)
41	2	nbusgreledg 27623	. . . . . . . . . . . . . . . 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 3899	. . . . . . . . . . . . . . . 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 4666	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
52	51	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
53	52	riotabidv 7214	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑚 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
54	53	cbvmptv 5183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
55	10, 54	eqtri 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 = (𝑚 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑚, 𝑦} ∈ 𝐸))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 55	frgrncvvdeqlem5 28568	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐷) → {(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
57		eleq2 2827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴‘𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴‘𝑚)}))
58	57	eqcoms 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({(𝐴‘𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴‘𝑚)}))
59		elsni 4575	. . . . . . . . . . . . . . . . . . . 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 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 234	. . . . . . . . . . . . . . 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 1921	. . . . 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 3069	. . . 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 3107	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚))
79		dffo3 6960	. 2 ⊢ (𝐴:𝐷–onto→𝑁 ↔ (𝐴:𝐷⟶𝑁 ∧ ∀𝑛 ∈ 𝑁 ∃𝑚 ∈ 𝐷 𝑛 = (𝐴‘𝑚)))
80	11, 78, 79	sylanbrc 582	1 ⊢ (𝜑 → 𝐴:𝐷–onto→𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065   ∩ cin 3882  {csn 4558  {cpr 4560   ↦ cmpt 5153  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Vtxcvtx 27269  Edgcedg 27320  USGraphcusgr 27422   NeighbVtx cnbgr 27602   FriendGraph cfrgr 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-edg 27321  df-upgr 27355  df-umgr 27356  df-usgr 27424  df-nbgr 27603  df-frgr 28524

This theorem is referenced by:  frgrncvvdeqlem10  28573