Theorem frgrancvvdeqlemB 26565

Description: Lemma B for frgrancvvdeq 26569. 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.)

Hypotheses

Ref	Expression
frgrancvvdeq.nx	⊢ 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
frgrancvvdeq.ny	⊢ 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
frgrancvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrancvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrancvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrancvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrancvvdeq.f	⊢ (𝜑 → 𝑉 FriendGrph 𝐸)
frgrancvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸))

Assertion

Ref	Expression
frgrancvvdeqlemB	⊢ (𝜑 → 𝐴:𝐷–1-1→ran 𝐴)

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

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

Proof of Theorem frgrancvvdeqlemB

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrancvvdeq.nx	. . 3 ⊢ 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
2		frgrancvvdeq.ny	. . 3 ⊢ 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
3		frgrancvvdeq.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		frgrancvvdeq.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5		frgrancvvdeq.ne	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
6		frgrancvvdeq.xy	. . 3 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
7		frgrancvvdeq.f	. . 3 ⊢ (𝜑 → 𝑉 FriendGrph 𝐸)
8		frgrancvvdeq.a	. . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
9	1, 2, 3, 4, 5, 6, 7, 8	frgrancvvdeqlem5 26561	. 2 ⊢ (𝜑 → 𝐴:𝐷⟶𝑁)
10		ffn 5958	. . . . . 6 ⊢ (𝐴:𝐷⟶𝑁 → 𝐴 Fn 𝐷)
11		dffn3 5967	. . . . . 6 ⊢ (𝐴 Fn 𝐷 ↔ 𝐴:𝐷⟶ran 𝐴)
12	10, 11	sylib 207	. . . . 5 ⊢ (𝐴:𝐷⟶𝑁 → 𝐴:𝐷⟶ran 𝐴)
13	12	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → 𝐴:𝐷⟶ran 𝐴)
14		ffvelrn 6265	. . . . . . . . . . . 12 ⊢ ((𝐴:𝐷⟶𝑁 ∧ 𝑢 ∈ 𝐷) → (𝐴‘𝑢) ∈ 𝑁)
15	14	adantll 746	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) → (𝐴‘𝑢) ∈ 𝑁)
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) → (𝐴‘𝑢) ∈ 𝑁)
17	16	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (𝐴‘𝑢) ∈ 𝑁)
18	1, 2, 3, 4, 5, 6, 7, 8	frgrancvvdeqlem2 26558	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
19		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
20	19	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ ran 𝐸 ↔ {𝑢, 𝑦} ∈ ran 𝐸))
21	20	riotabidv 6513	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑢 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸) = (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ ran 𝐸))
22	21	cbvmptv 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) = (𝑢 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ ran 𝐸))
23	8, 22	eqtri 2632	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐴 = (𝑢 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ ran 𝐸))
24	1, 2, 3, 4, 5, 6, 7, 23	frgrancvvdeqlem7 26563	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐷) → {𝑢, (𝐴‘𝑢)} ∈ ran 𝐸)
25		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
26	25	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ ran 𝐸 ↔ {𝑤, 𝑦} ∈ ran 𝐸))
27	26	riotabidv 6513	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑤 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸) = (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ ran 𝐸))
28	27	cbvmptv 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) = (𝑤 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ ran 𝐸))
29	8, 28	eqtri 2632	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐴 = (𝑤 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ ran 𝐸))
30	1, 2, 3, 4, 5, 6, 7, 29	frgrancvvdeqlem7 26563	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸)
31	24, 30	anim12dan 878	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸))
32		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → {𝑤, (𝐴‘𝑤)} = {𝑤, (𝐴‘𝑢)})
33	32	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → ({𝑤, (𝐴‘𝑤)} ∈ ran 𝐸 ↔ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸))
34	33	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸) ↔ ({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸)))
35	34	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴‘𝑢) = (𝐴‘𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸) ↔ ({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸)))
36	35	biimpa 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴‘𝑢) = (𝐴‘𝑤) ∧ ({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸)) → ({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸))
37		df-ne 2782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ≠ 𝑤 ↔ ¬ 𝑢 = 𝑤)
38	3, 1, 7	frgranbnb 26547	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸) → (𝐴‘𝑢) = 𝑋))
39	38	3expa 1257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸) → (𝐴‘𝑢) = 𝑋))
40		df-nel 2783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
41		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 ↔ 𝑋 ∈ 𝑁))
42	41	biimpa 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐴‘𝑢) = 𝑋 ∧ (𝐴‘𝑢) ∈ 𝑁) → 𝑋 ∈ 𝑁)
43	42	pm2.24d 146	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐴‘𝑢) = 𝑋 ∧ (𝐴‘𝑢) ∈ 𝑁) → (¬ 𝑋 ∈ 𝑁 → 𝑢 = 𝑤))
44	43	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴‘𝑢) ∈ 𝑁 → ((𝐴‘𝑢) = 𝑋 → (¬ 𝑋 ∈ 𝑁 → 𝑢 = 𝑤)))
45	44	com13 86	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
46	40, 45	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
47	46	com12 32	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴‘𝑢) = 𝑋 → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
48	39, 47	syl6 34	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))
49	48	expcom 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 ≠ 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
50	49	com23 84	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ≠ 𝑤 → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
51	37, 50	sylbir 224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑢 = 𝑤 → (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ ran 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
52	36, 51	syl5com 31	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴‘𝑢) = (𝐴‘𝑤) ∧ ({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸)) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
53	52	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
54	53	com24 93	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑢, (𝐴‘𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ ran 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
55	31, 54	mpcom 37	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
56	55	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
57	56	com3r 85	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑢 = 𝑤 → (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
58	57	com15 99	. . . . . . . . . . . . 13 ⊢ (𝑋 ∉ 𝑁 → (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
59	18, 58	mpcom 37	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))
60	59	expd 451	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ 𝐷 → (𝑤 ∈ 𝐷 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → (𝑢 ∈ 𝐷 → (𝑤 ∈ 𝐷 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))))
62	61	imp41 617	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))
63	17, 62	mpid 43	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (¬ 𝑢 = 𝑤 → 𝑢 = 𝑤))
64	63	pm2.18d 123	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → 𝑢 = 𝑤)
65	64	ex 449	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤))
66	65	ralrimiva 2949	. . . . 5 ⊢ (((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) → ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤))
67	66	ralrimiva 2949	. . . 4 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → ∀𝑢 ∈ 𝐷 ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤))
68		dff13 6416	. . . 4 ⊢ (𝐴:𝐷–1-1→ran 𝐴 ↔ (𝐴:𝐷⟶ran 𝐴 ∧ ∀𝑢 ∈ 𝐷 ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤)))
69	13, 67, 68	sylanbrc 695	. . 3 ⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → 𝐴:𝐷–1-1→ran 𝐴)
70	69	expcom 450	. 2 ⊢ (𝐴:𝐷⟶𝑁 → (𝜑 → 𝐴:𝐷–1-1→ran 𝐴))
71	9, 70	mpcom 37	1 ⊢ (𝜑 → 𝐴:𝐷–1-1→ran 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   Neighbors cnbgra 25946   FriendGrph cfrgra 26515

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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-usgra 25862  df-nbgra 25949  df-frgra 26516

This theorem is referenced by:  frgrancvvdeqlem8  26567