Theorem frgrncvvdeqlem8 30399

Description: Lemma 8 for frgrncvvdeq 30402. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Revised by AV, 30-Dec-2021.)

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
frgrncvvdeqlem8	⊢ (𝜑 → 𝐴:𝐷–1-1→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 frgrncvvdeqlem8

Dummy variables 𝑢 𝑤 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 30395	. 2 ⊢ (𝜑 → 𝐴:𝐷⟶𝑁)
12		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → 𝐴:𝐷⟶𝑁)
13		ffvelcdm 7037	. . . . . . . . 9 ⊢ ((𝐴:𝐷⟶𝑁 ∧ 𝑢 ∈ 𝐷) → (𝐴‘𝑢) ∈ 𝑁)
14	13	ad2ant2lr 749	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝐴‘𝑢) ∈ 𝑁)
15	14	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (𝐴‘𝑢) ∈ 𝑁)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem1 30392	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
17		preq1 4692	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
18	17	eleq1d 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑢, 𝑦} ∈ 𝐸))
19	18	riotabidv 7329	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑢 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ 𝐸))
20	19	cbvmptv 5204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑢 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ 𝐸))
21	10, 20	eqtri 2760	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 = (𝑢 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ 𝐸))
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 21	frgrncvvdeqlem6 30397	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐷) → {𝑢, (𝐴‘𝑢)} ∈ 𝐸)
23		preq1 4692	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
24	23	eleq1d 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑤, 𝑦} ∈ 𝐸))
25	24	riotabidv 7329	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑤 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ 𝐸))
26	25	cbvmptv 5204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑤 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ 𝐸))
27	10, 26	eqtri 2760	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 = (𝑤 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ 𝐸))
28	1, 2, 3, 4, 5, 6, 7, 8, 9, 27	frgrncvvdeqlem6 30397	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → {𝑤, (𝐴‘𝑤)} ∈ 𝐸)
29	22, 28	anim12dan 620	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸))
30		preq2 4693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → {𝑤, (𝐴‘𝑤)} = {𝑤, (𝐴‘𝑢)})
31	30	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → ({𝑤, (𝐴‘𝑤)} ∈ 𝐸 ↔ {𝑤, (𝐴‘𝑢)} ∈ 𝐸))
32	31	anbi2d 631	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸)))
33	32	eqcoms 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑢) = (𝐴‘𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸)))
34	33	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑢) = (𝐴‘𝑤) ∧ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸)) → ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸))
35		df-ne 2934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ≠ 𝑤 ↔ ¬ 𝑢 = 𝑤)
36	2, 3	frgrnbnb 30386	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝐴‘𝑢) = 𝑋))
37	9, 36	syl3an1 1164	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝐴‘𝑢) = 𝑋))
38	37	3expa 1119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝐴‘𝑢) = 𝑋))
39		df-nel 3038	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
40		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 ↔ 𝑋 ∈ 𝑁))
41	40	biimpa 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐴‘𝑢) = 𝑋 ∧ (𝐴‘𝑢) ∈ 𝑁) → 𝑋 ∈ 𝑁)
42	41	pm2.24d 151	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐴‘𝑢) = 𝑋 ∧ (𝐴‘𝑢) ∈ 𝑁) → (¬ 𝑋 ∈ 𝑁 → 𝑢 = 𝑤))
43	42	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴‘𝑢) ∈ 𝑁 → ((𝐴‘𝑢) = 𝑋 → (¬ 𝑋 ∈ 𝑁 → 𝑢 = 𝑤)))
44	43	com13 88	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
45	39, 44	sylbi 217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
46	45	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴‘𝑢) = 𝑋 → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
47	38, 46	syl6 35	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))
48	47	expcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ≠ 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
49	48	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ≠ 𝑤 → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
50	35, 49	sylbir 235	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑢 = 𝑤 → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
51	34, 50	syl5com 31	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑢) = (𝐴‘𝑤) ∧ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸)) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
52	51	expcom 413	. . . . . . . . . . . . . . . 16 ⊢ (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
53	52	com24 95	. . . . . . . . . . . . . . 15 ⊢ (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
54	29, 53	mpcom 38	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
55	54	ex 412	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
56	55	com3r 87	. . . . . . . . . . . 12 ⊢ (¬ 𝑢 = 𝑤 → (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
57	56	com15 101	. . . . . . . . . . 11 ⊢ (𝑋 ∉ 𝑁 → (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
58	16, 57	mpcom 38	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
59	58	expd 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑢 ∈ 𝐷 → (𝑤 ∈ 𝐷 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
60	59	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → (𝑢 ∈ 𝐷 → (𝑤 ∈ 𝐷 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
61	60	imp42 426	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
62	15, 61	mpid 44	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (¬ 𝑢 = 𝑤 → 𝑢 = 𝑤))
63	62	pm2.18d 127	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → 𝑢 = 𝑤)
64	63	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤))
65	64	ralrimivva 3181	. . 3 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → ∀𝑢 ∈ 𝐷 ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤))
66		dff13 7212	. . 3 ⊢ (𝐴:𝐷–1-1→𝑁 ↔ (𝐴:𝐷⟶𝑁 ∧ ∀𝑢 ∈ 𝐷 ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤)))
67	12, 65, 66	sylanbrc 584	. 2 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → 𝐴:𝐷–1-1→𝑁)
68	11, 67	mpdan 688	1 ⊢ (𝜑 → 𝐴:𝐷–1-1→𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052  {cpr 4584   ↦ cmpt 5181  ⟶wf 6498  –1-1→wf1 6499  ‘cfv 6502  ℩crio 7326  (class class class)co 7370  Vtxcvtx 29087  Edgcedg 29138   NeighbVtx cnbgr 29423   FriendGraph cfrgr 30351

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 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-xnn0 12489  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268  df-edg 29139  df-upgr 29173  df-umgr 29174  df-usgr 29242  df-nbgr 29424  df-frgr 30352

This theorem is referenced by:  frgrncvvdeqlem10  30401