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Theorem frgrancvvdeqlemB 26327
Description: Lemma B for frgrancvvdeq 26331. 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 (𝜑𝑋𝑉)
frgrancvvdeq.y (𝜑𝑌𝑉)
frgrancvvdeq.ne (𝜑𝑋𝑌)
frgrancvvdeq.xy (𝜑𝑌𝐷)
frgrancvvdeq.f (𝜑𝑉 FriendGrph 𝐸)
frgrancvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
Assertion
Ref Expression
frgrancvvdeqlemB (𝜑𝐴:𝐷1-1→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.
StepHypRef Expression
1 frgrancvvdeq.nx . . 3 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
2 frgrancvvdeq.ny . . 3 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
3 frgrancvvdeq.x . . 3 (𝜑𝑋𝑉)
4 frgrancvvdeq.y . . 3 (𝜑𝑌𝑉)
5 frgrancvvdeq.ne . . 3 (𝜑𝑋𝑌)
6 frgrancvvdeq.xy . . 3 (𝜑𝑌𝐷)
7 frgrancvvdeq.f . . 3 (𝜑𝑉 FriendGrph 𝐸)
8 frgrancvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem5 26323 . 2 (𝜑𝐴:𝐷𝑁)
10 ffn 5940 . . . . . 6 (𝐴:𝐷𝑁𝐴 Fn 𝐷)
11 dffn3 5949 . . . . . 6 (𝐴 Fn 𝐷𝐴:𝐷⟶ran 𝐴)
1210, 11sylib 206 . . . . 5 (𝐴:𝐷𝑁𝐴:𝐷⟶ran 𝐴)
1312adantl 480 . . . 4 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷⟶ran 𝐴)
14 ffvelrn 6246 . . . . . . . . . . . 12 ((𝐴:𝐷𝑁𝑢𝐷) → (𝐴𝑢) ∈ 𝑁)
1514adantll 745 . . . . . . . . . . 11 (((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) → (𝐴𝑢) ∈ 𝑁)
1615adantr 479 . . . . . . . . . 10 ((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) → (𝐴𝑢) ∈ 𝑁)
1716adantr 479 . . . . . . . . 9 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → (𝐴𝑢) ∈ 𝑁)
181, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 26320 . . . . . . . . . . . . 13 (𝜑𝑋𝑁)
19 preq1 4207 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
2019eleq1d 2667 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ ran 𝐸 ↔ {𝑢, 𝑦} ∈ ran 𝐸))
2120riotabidv 6487 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑢 → (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸) = (𝑦𝑁 {𝑢, 𝑦} ∈ ran 𝐸))
2221cbvmptv 4668 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ ran 𝐸))
238, 22eqtri 2627 . . . . . . . . . . . . . . . . . . 19 𝐴 = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ ran 𝐸))
241, 2, 3, 4, 5, 6, 7, 23frgrancvvdeqlem7 26325 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑢𝐷) → {𝑢, (𝐴𝑢)} ∈ ran 𝐸)
25 preq1 4207 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
2625eleq1d 2667 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ ran 𝐸 ↔ {𝑤, 𝑦} ∈ ran 𝐸))
2726riotabidv 6487 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸) = (𝑦𝑁 {𝑤, 𝑦} ∈ ran 𝐸))
2827cbvmptv 4668 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ ran 𝐸))
298, 28eqtri 2627 . . . . . . . . . . . . . . . . . . 19 𝐴 = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ ran 𝐸))
301, 2, 3, 4, 5, 6, 7, 29frgrancvvdeqlem7 26325 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤𝐷) → {𝑤, (𝐴𝑤)} ∈ ran 𝐸)
3124, 30anim12dan 877 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → ({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸))
32 preq2 4208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑤) = (𝐴𝑢) → {𝑤, (𝐴𝑤)} = {𝑤, (𝐴𝑢)})
3332eleq1d 2667 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑤) = (𝐴𝑢) → ({𝑤, (𝐴𝑤)} ∈ ran 𝐸 ↔ {𝑤, (𝐴𝑢)} ∈ ran 𝐸))
3433anbi2d 735 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑤) = (𝐴𝑢) → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸)))
3534eqcoms 2613 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑢) = (𝐴𝑤) → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸)))
3635biimpa 499 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸)) → ({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸))
37 df-ne 2777 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝑤 ↔ ¬ 𝑢 = 𝑤)
383, 1, 7frgranbnb 26309 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸) → (𝐴𝑢) = 𝑋))
39383expa 1256 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸) → (𝐴𝑢) = 𝑋))
40 df-nel 2778 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
41 eleq1 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑋𝑁))
4241biimpa 499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → 𝑋𝑁)
4342pm2.24d 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → (¬ 𝑋𝑁𝑢 = 𝑤))
4443expcom 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴𝑢) ∈ 𝑁 → ((𝐴𝑢) = 𝑋 → (¬ 𝑋𝑁𝑢 = 𝑤)))
4544com13 85 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4640, 45sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4746com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑢) = 𝑋 → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4839, 47syl6 34 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))
4948expcom 449 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5049com23 83 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝑤 → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5137, 50sylbir 223 . . . . . . . . . . . . . . . . . . . 20 𝑢 = 𝑤 → (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ ran 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5236, 51syl5com 31 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸)) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5352expcom 449 . . . . . . . . . . . . . . . . . 18 (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5453com24 92 . . . . . . . . . . . . . . . . 17 (({𝑢, (𝐴𝑢)} ∈ ran 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ ran 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5531, 54mpcom 37 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5655ex 448 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑢𝐷𝑤𝐷) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5756com3r 84 . . . . . . . . . . . . . 14 𝑢 = 𝑤 → (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5857com15 98 . . . . . . . . . . . . 13 (𝑋𝑁 → (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5918, 58mpcom 37 . . . . . . . . . . . 12 (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
6059expd 450 . . . . . . . . . . 11 (𝜑 → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6160adantr 479 . . . . . . . . . 10 ((𝜑𝐴:𝐷𝑁) → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6261imp41 616 . . . . . . . . 9 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
6317, 62mpid 42 . . . . . . . 8 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → (¬ 𝑢 = 𝑤𝑢 = 𝑤))
6463pm2.18d 122 . . . . . . 7 (((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) ∧ (𝐴𝑢) = (𝐴𝑤)) → 𝑢 = 𝑤)
6564ex 448 . . . . . 6 ((((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) ∧ 𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
6665ralrimiva 2944 . . . . 5 (((𝜑𝐴:𝐷𝑁) ∧ 𝑢𝐷) → ∀𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
6766ralrimiva 2944 . . . 4 ((𝜑𝐴:𝐷𝑁) → ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
68 dff13 6390 . . . 4 (𝐴:𝐷1-1→ran 𝐴 ↔ (𝐴:𝐷⟶ran 𝐴 ∧ ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤)))
6913, 67, 68sylanbrc 694 . . 3 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷1-1→ran 𝐴)
7069expcom 449 . 2 (𝐴:𝐷𝑁 → (𝜑𝐴:𝐷1-1→ran 𝐴))
719, 70mpcom 37 1 (𝜑𝐴:𝐷1-1→ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wne 2775  wnel 2776  wral 2891  {cpr 4122  cop 4126   class class class wbr 4573  cmpt 4633  ran crn 5025   Fn wfn 5781  wf 5782  1-1wf1 5783  cfv 5786  crio 6484  (class class class)co 6523   Neighbors cnbgra 25708   FriendGrph cfrgra 26277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-hash 12931  df-usgra 25624  df-nbgra 25711  df-frgra 26278
This theorem is referenced by:  frgrancvvdeqlem8  26329
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