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Theorem frgrncvvdeqlem9 28000
 Description: Lemma 9 for frgrncvvdeq 28002. 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 (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
frgrncvvdeqlem9 (𝜑𝐴:𝐷onto𝑁)
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.
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, 10frgrncvvdeqlem4 27995 . 2 (𝜑𝐴:𝐷𝑁)
129adantr 481 . . . . . . 7 ((𝜑𝑛𝑁) → 𝐺 ∈ FriendGraph )
134eleq2i 2909 . . . . . . . . . 10 (𝑛𝑁𝑛 ∈ (𝐺 NeighbVtx 𝑌))
141nbgrisvtx 27037 . . . . . . . . . . 11 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉)
1514a1i 11 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
1613, 15syl5bi 243 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑉))
1716imp 407 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑉)
185adantr 481 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑋𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem1 27992 . . . . . . . . . 10 (𝜑𝑋𝑁)
20 df-nel 3129 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
21 nelelne 3122 . . . . . . . . . . 11 𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2220, 21sylbi 218 . . . . . . . . . 10 (𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2319, 22syl 17 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑋))
2423imp 407 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑋)
2517, 18, 243jca 1122 . . . . . . 7 ((𝜑𝑛𝑁) → (𝑛𝑉𝑋𝑉𝑛𝑋))
2612, 25jca 512 . . . . . 6 ((𝜑𝑛𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)))
271, 2frcond2 27960 . . . . . . 7 (𝐺 ∈ FriendGraph → ((𝑛𝑉𝑋𝑉𝑛𝑋) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
2827imp 407 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
29 reurex 3437 . . . . . . 7 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
30 df-rex 3149 . . . . . . 7 (∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3129, 30sylib 219 . . . . . 6 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3226, 28, 313syl 18 . . . . 5 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
33 frgrusgr 27954 . . . . . . . . . . . . 13 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
342nbusgreledg 27049 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
3534bicomd 224 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
369, 33, 353syl 18 . . . . . . . . . . . 12 (𝜑 → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
3736biimpa 477 . . . . . . . . . . 11 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
383eleq2i 2909 . . . . . . . . . . 11 (𝑚𝐷𝑚 ∈ (𝐺 NeighbVtx 𝑋))
3937, 38sylibr 235 . . . . . . . . . 10 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚𝐷)
4039ad2ant2rl 745 . . . . . . . . 9 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑚𝐷)
412nbusgreledg 27049 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ↔ {𝑛, 𝑚} ∈ 𝐸))
4241biimpar 478 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝑛, 𝑚} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
4342a1d 25 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ {𝑛, 𝑚} ∈ 𝐸) → ({𝑚, 𝑋} ∈ 𝐸𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4443expimpd 454 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
459, 33, 443syl 18 . . . . . . . . . . . 12 (𝜑 → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4645adantr 481 . . . . . . . . . . 11 ((𝜑𝑛𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4746imp 407 . . . . . . . . . 10 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
48 elin 4173 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁))
49 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝜑)
5049, 39jca 512 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝜑𝑚𝐷))
51 preq1 4668 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
5251eleq1d 2902 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
5352riotabidv 7108 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑚 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
5453cbvmptv 5166 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
5510, 54eqtri 2849 . . . . . . . . . . . . . . . . . . . 20 𝐴 = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
561, 2, 3, 4, 5, 6, 7, 8, 9, 55frgrncvvdeqlem5 27996 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝐷) → {(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
57 eleq2 2906 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
5857eqcoms 2834 . . . . . . . . . . . . . . . . . . . 20 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
59 elsni 4581 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ {(𝐴𝑚)} → 𝑛 = (𝐴𝑚))
6058, 59syl6bi 254 . . . . . . . . . . . . . . . . . . 19 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚)))
6150, 56, 603syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚)))
6261expcom 414 . . . . . . . . . . . . . . . . 17 ({𝑚, 𝑋} ∈ 𝐸 → (𝜑 → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
6362com3r 87 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑𝑛 = (𝐴𝑚))))
6448, 63sylbir 236 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑𝑛 = (𝐴𝑚))))
6564ex 413 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → (𝑛𝑁 → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑𝑛 = (𝐴𝑚)))))
6665com14 96 . . . . . . . . . . . . 13 (𝜑 → (𝑛𝑁 → ({𝑚, 𝑋} ∈ 𝐸 → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚)))))
6766imp 407 . . . . . . . . . . . 12 ((𝜑𝑛𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚))))
6867adantld 491 . . . . . . . . . . 11 ((𝜑𝑛𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚))))
6968imp 407 . . . . . . . . . 10 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚)))
7047, 69mpd 15 . . . . . . . . 9 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 = (𝐴𝑚))
7140, 70jca 512 . . . . . . . 8 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))
7271ex 413 . . . . . . 7 ((𝜑𝑛𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑚𝐷𝑛 = (𝐴𝑚))))
7372adantld 491 . . . . . 6 ((𝜑𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚))))
7473eximdv 1911 . . . . 5 ((𝜑𝑛𝑁) → (∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚))))
7532, 74mpd 15 . . . 4 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
76 df-rex 3149 . . . 4 (∃𝑚𝐷 𝑛 = (𝐴𝑚) ↔ ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
7775, 76sylibr 235 . . 3 ((𝜑𝑛𝑁) → ∃𝑚𝐷 𝑛 = (𝐴𝑚))
7877ralrimiva 3187 . 2 (𝜑 → ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚))
79 dffo3 6864 . 2 (𝐴:𝐷onto𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚)))
8011, 78, 79sylanbrc 583 1 (𝜑𝐴:𝐷onto𝑁)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530  ∃wex 1773   ∈ wcel 2107   ≠ wne 3021   ∉ wnel 3128  ∀wral 3143  ∃wrex 3144  ∃!wreu 3145   ∩ cin 3939  {csn 4564  {cpr 4566   ↦ cmpt 5143  ⟶wf 6348  –onto→wfo 6350  ‘cfv 6352  ℩crio 7105  (class class class)co 7148  Vtxcvtx 26695  Edgcedg 26746  USGraphcusgr 26848   NeighbVtx cnbgr 27028   FriendGraph cfrgr 27951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681  df-edg 26747  df-upgr 26781  df-umgr 26782  df-usgr 26850  df-nbgr 27029  df-frgr 27952 This theorem is referenced by:  frgrncvvdeqlem10  28001
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