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Theorem frgrncvvdeqlem9 27287
 Description: Lemma 9 for frgrncvvdeq 27289. 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,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦,𝐸   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 27282 . 2 (𝜑𝐴:𝐷𝑁)
129adantr 480 . . . . . . 7 ((𝜑𝑛𝑁) → 𝐺 ∈ FriendGraph )
134eleq2i 2722 . . . . . . . . . 10 (𝑛𝑁𝑛 ∈ (𝐺 NeighbVtx 𝑌))
141nbgrisvtx 26280 . . . . . . . . . . 11 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉)
1514a1i 11 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
1613, 15syl5bi 232 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑉))
1716imp 444 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑉)
185adantr 480 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑋𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem1 27279 . . . . . . . . . 10 (𝜑𝑋𝑁)
20 df-nel 2927 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
21 nelelne 2921 . . . . . . . . . . 11 𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2220, 21sylbi 207 . . . . . . . . . 10 (𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2319, 22syl 17 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑋))
2423imp 444 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑋)
2517, 18, 243jca 1261 . . . . . . 7 ((𝜑𝑛𝑁) → (𝑛𝑉𝑋𝑉𝑛𝑋))
2612, 25jca 553 . . . . . 6 ((𝜑𝑛𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)))
271, 2frcond2 27247 . . . . . . 7 (𝐺 ∈ FriendGraph → ((𝑛𝑉𝑋𝑉𝑛𝑋) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
2827imp 444 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
29 reurex 3190 . . . . . . 7 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
30 df-rex 2947 . . . . . . 7 (∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3129, 30sylib 208 . . . . . 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 27240 . . . . . . . . . . . . 13 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
342nbusgreledg 26294 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
3534bicomd 213 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
369, 33, 353syl 18 . . . . . . . . . . . 12 (𝜑 → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
3736biimpa 500 . . . . . . . . . . 11 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
383eleq2i 2722 . . . . . . . . . . 11 (𝑚𝐷𝑚 ∈ (𝐺 NeighbVtx 𝑋))
3937, 38sylibr 224 . . . . . . . . . 10 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚𝐷)
4039ad2ant2rl 800 . . . . . . . . 9 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑚𝐷)
412nbusgreledg 26294 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ↔ {𝑛, 𝑚} ∈ 𝐸))
4241biimpar 501 . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑛𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4746imp 444 . . . . . . . . . 10 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
48 elin 3829 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁))
49 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝜑)
5049, 39jca 553 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝜑𝑚𝐷))
51 preq1 4300 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
5251eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
5352riotabidv 6653 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑚 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
5453cbvmptv 4783 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
5510, 54eqtri 2673 . . . . . . . . . . . . . . . . . . . 20 𝐴 = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
561, 2, 3, 4, 5, 6, 7, 8, 9, 55frgrncvvdeqlem5 27283 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝐷) → {(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
57 eleq2 2719 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
5857eqcoms 2659 . . . . . . . . . . . . . . . . . . . 20 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
59 elsni 4227 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ {(𝐴𝑚)} → 𝑛 = (𝐴𝑚))
6058, 59syl6bi 243 . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝜑𝑛 = (𝐴𝑚))))
6564ex 449 . . . . . . . . . . . . . 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 444 . . . . . . . . . . . 12 ((𝜑𝑛𝑁) → ({𝑚, 𝑋} ∈ 𝐸 → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚))))
6867adantld 482 . . . . . . . . . . 11 ((𝜑𝑛𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚))))
6968imp 444 . . . . . . . . . 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 553 . . . . . . . 8 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))
7271ex 449 . . . . . . 7 ((𝜑𝑛𝑁) → (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑚𝐷𝑛 = (𝐴𝑚))))
7372adantld 482 . . . . . 6 ((𝜑𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚))))
7473eximdv 1886 . . . . 5 ((𝜑𝑛𝑁) → (∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚))))
7532, 74mpd 15 . . . 4 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
76 df-rex 2947 . . . 4 (∃𝑚𝐷 𝑛 = (𝐴𝑚) ↔ ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
7775, 76sylibr 224 . . 3 ((𝜑𝑛𝑁) → ∃𝑚𝐷 𝑛 = (𝐴𝑚))
7877ralrimiva 2995 . 2 (𝜑 → ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚))
79 dffo3 6414 . 2 (𝐴:𝐷onto𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚)))
8011, 78, 79sylanbrc 699 1 (𝜑𝐴:𝐷onto𝑁)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823   ∉ wnel 2926  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943   ∩ cin 3606  {csn 4210  {cpr 4212   ↦ cmpt 4762  ⟶wf 5922  –onto→wfo 5924  ‘cfv 5926  ℩crio 6650  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   NeighbVtx cnbgr 26269   FriendGraph cfrgr 27236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-upgr 26022  df-umgr 26023  df-usgr 26091  df-nbgr 26270  df-frgr 27237 This theorem is referenced by:  frgrncvvdeqlem10  27288
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