Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrncvvdeqlemB Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlemB 41477
 Description: Lemma B for frgrncvvdeq 41480. 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.)
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 (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlemB (𝜑𝐴:𝐷1-1→ran 𝐴)
Distinct variable groups:   𝑦,𝐷   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝜑,𝑦,𝑥   𝑦,𝐸   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥   𝑥,𝐸
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlemB
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef 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 (𝜑𝑋𝑉)
6 frgrncvvdeq.y . . 3 (𝜑𝑌𝑉)
7 frgrncvvdeq.ne . . 3 (𝜑𝑋𝑌)
8 frgrncvvdeq.xy . . 3 (𝜑𝑌𝐷)
9 frgrncvvdeq.f . . 3 (𝜑𝐺 ∈ FriendGraph )
10 frgrncvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem5 41473 . 2 (𝜑𝐴:𝐷𝑁)
12 ffrn 5968 . . . . 5 (𝐴:𝐷𝑁𝐴:𝐷⟶ran 𝐴)
1312adantl 481 . . . 4 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷⟶ran 𝐴)
14 ffvelrn 6265 . . . . . . . . . 10 ((𝐴:𝐷𝑁𝑢𝐷) → (𝐴𝑢) ∈ 𝑁)
1514ad5ant23 1296 . . . . . . . . 9 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → (𝐴𝑢) ∈ 𝑁)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem2 41470 . . . . . . . . . . . . 13 (𝜑𝑋𝑁)
17 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
1817eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑢, 𝑦} ∈ 𝐸))
1918riotabidv 6513 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑢 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2019cbvmptv 4678 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2110, 20eqtri 2632 . . . . . . . . . . . . . . . . . . 19 𝐴 = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
221, 2, 3, 4, 5, 6, 7, 8, 9, 21frgrncvvdeqlem7 41475 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢𝐷) → {𝑢, (𝐴𝑢)} ∈ 𝐸)
23 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
2423eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑤, 𝑦} ∈ 𝐸))
2524riotabidv 6513 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2625cbvmptv 4678 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2710, 26eqtri 2632 . . . . . . . . . . . . . . . . . . 19 𝐴 = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
281, 2, 3, 4, 5, 6, 7, 8, 9, 27frgrncvvdeqlem7 41475 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝐷) → {𝑤, (𝐴𝑤)} ∈ 𝐸)
2922, 28anim12dan 878 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸))
30 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑤) = (𝐴𝑢) → {𝑤, (𝐴𝑤)} = {𝑤, (𝐴𝑢)})
3130eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑤) = (𝐴𝑢) → ({𝑤, (𝐴𝑤)} ∈ 𝐸 ↔ {𝑤, (𝐴𝑢)} ∈ 𝐸))
3231anbi2d 736 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑤) = (𝐴𝑢) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸)))
3332eqcoms 2618 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑢) = (𝐴𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸)))
3433biimpa 500 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸))
35 df-ne 2782 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝑤 ↔ ¬ 𝑢 = 𝑤)
362, 3frgrnbnb 41463 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ FriendGraph ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
379, 36syl3an1 1351 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
38373expa 1257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
39 df-nel 2783 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
40 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑋𝑁))
4140biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → 𝑋𝑁)
4241pm2.24d 146 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → (¬ 𝑋𝑁𝑢 = 𝑤))
4342expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝑢) ∈ 𝑁 → ((𝐴𝑢) = 𝑋 → (¬ 𝑋𝑁𝑢 = 𝑤)))
4443com13 86 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4539, 44sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4645com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑢) = 𝑋 → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4738, 46syl6 34 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))
4847expcom 450 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
4948com23 84 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝑤 → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5035, 49sylbir 224 . . . . . . . . . . . . . . . . . . . 20 𝑢 = 𝑤 → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5134, 50syl5com 31 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸)) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5251expcom 450 . . . . . . . . . . . . . . . . . 18 (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5352com24 93 . . . . . . . . . . . . . . . . 17 (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5429, 53mpcom 37 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5554ex 449 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑢𝐷𝑤𝐷) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5655com3r 85 . . . . . . . . . . . . . 14 𝑢 = 𝑤 → (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5756com15 99 . . . . . . . . . . . . 13 (𝑋𝑁 → (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5816, 57mpcom 37 . . . . . . . . . . . 12 (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5958expd 451 . . . . . . . . . . 11 (𝜑 → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6059adantr 480 . . . . . . . . . 10 ((𝜑𝐴:𝐷𝑁) → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6160imp41 617 . . . . . . . . 9 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
6215, 61mpid 43 . . . . . . . 8 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → (¬ 𝑢 = 𝑤𝑢 = 𝑤))
6362pm2.18d 123 . . . . . . 7 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → 𝑢 = 𝑤)
6463ex 449 . . . . . 6 ((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
6564ralrimiva 2949 . . . . 5 (((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) → ∀𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
6665ralrimiva 2949 . . . 4 ((𝜑𝐴:𝐷𝑁) → ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
67 dff13 6416 . . . 4 (𝐴:𝐷1-1→ran 𝐴 ↔ (𝐴:𝐷⟶ran 𝐴 ∧ ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤)))
6813, 66, 67sylanbrc 695 . . 3 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷1-1→ran 𝐴)
6968expcom 450 . 2 (𝐴:𝐷𝑁 → (𝜑𝐴:𝐷1-1→ran 𝐴))
7011, 69mpcom 37 1 (𝜑𝐴:𝐷1-1→ran 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {cpr 4127   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   NeighbVtx cnbgr 40550   FriendGraph cfrgr 41428 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-rep 4699  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-fal 1481  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-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  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-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-frgr 41429 This theorem is referenced by:  frgrncvvdeqlem8  41479
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