MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrncvvdeqlem9 Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlem9 29251
Description: Lemma 9 for frgrncvvdeq 29253. 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 29246 . 2 (𝜑𝐴:𝐷𝑁)
129adantr 481 . . . . . . 7 ((𝜑𝑛𝑁) → 𝐺 ∈ FriendGraph )
134eleq2i 2829 . . . . . . . . . 10 (𝑛𝑁𝑛 ∈ (𝐺 NeighbVtx 𝑌))
141nbgrisvtx 28289 . . . . . . . . . . 11 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉)
1514a1i 11 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
1613, 15biimtrid 241 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑉))
1716imp 407 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑉)
185adantr 481 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑋𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem1 29243 . . . . . . . . . 10 (𝜑𝑋𝑁)
20 df-nel 3050 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
21 nelelne 3043 . . . . . . . . . . 11 𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2220, 21sylbi 216 . . . . . . . . . 10 (𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2319, 22syl 17 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑋))
2423imp 407 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑋)
2517, 18, 243jca 1128 . . . . . . 7 ((𝜑𝑛𝑁) → (𝑛𝑉𝑋𝑉𝑛𝑋))
2612, 25jca 512 . . . . . 6 ((𝜑𝑛𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)))
271, 2frcond2 29211 . . . . . . 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 3357 . . . . . . 7 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
30 df-rex 3074 . . . . . . 7 (∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3129, 30sylib 217 . . . . . 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 29205 . . . . . . . . . . . . 13 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
342nbusgreledg 28301 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
3534bicomd 222 . . . . . . . . . . . . 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 2829 . . . . . . . . . . 11 (𝑚𝐷𝑚 ∈ (𝐺 NeighbVtx 𝑋))
3937, 38sylibr 233 . . . . . . . . . 10 ((𝜑 ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚𝐷)
4039ad2ant2rl 747 . . . . . . . . 9 (((𝜑𝑛𝑁) ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑚𝐷)
412nbusgreledg 28301 . . . . . . . . . . . . . . . 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 3926 . . . . . . . . . . . . . . . 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 4694 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
5251eleq1d 2822 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
5352riotabidv 7315 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑚 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
5453cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
5510, 54eqtri 2764 . . . . . . . . . . . . . . . . . . . 20 𝐴 = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
561, 2, 3, 4, 5, 6, 7, 8, 9, 55frgrncvvdeqlem5 29247 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝐷) → {(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
57 eleq2 2826 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
5857eqcoms 2744 . . . . . . . . . . . . . . . . . . . 20 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
59 elsni 4603 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ {(𝐴𝑚)} → 𝑛 = (𝐴𝑚))
6058, 59syl6bi 252 . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . 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 1920 . . . . 5 ((𝜑𝑛𝑁) → (∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚))))
7532, 74mpd 15 . . . 4 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
76 df-rex 3074 . . . 4 (∃𝑚𝐷 𝑛 = (𝐴𝑚) ↔ ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
7775, 76sylibr 233 . . 3 ((𝜑𝑛𝑁) → ∃𝑚𝐷 𝑛 = (𝐴𝑚))
7877ralrimiva 3143 . 2 (𝜑 → ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚))
79 dffo3 7052 . 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wnel 3049  wral 3064  wrex 3073  ∃!wreu 3351  cin 3909  {csn 4586  {cpr 4588  cmpt 5188  wf 6492  ontowfo 6494  cfv 6496  crio 7312  (class class class)co 7357  Vtxcvtx 27947  Edgcedg 27998  USGraphcusgr 28100   NeighbVtx cnbgr 28280   FriendGraph cfrgr 29202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-upgr 28033  df-umgr 28034  df-usgr 28102  df-nbgr 28281  df-frgr 29203
This theorem is referenced by:  frgrncvvdeqlem10  29252
  Copyright terms: Public domain W3C validator