MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrncvvdeqlem3 Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlem3 29551
Description: Lemma 3 for frgrncvvdeq 29559. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-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
frgrncvvdeqlem3 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦   𝑦,𝑁
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem3
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 frgrncvvdeq.ny . . 3 𝑁 = (𝐺 NeighbVtx 𝑌)
21ineq2i 4209 . 2 ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))
3 frgrncvvdeq.f . . . . 5 (𝜑𝐺 ∈ FriendGraph )
43adantr 481 . . . 4 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
5 frgrncvvdeq.nx . . . . . . . 8 𝐷 = (𝐺 NeighbVtx 𝑋)
65eleq2i 2825 . . . . . . 7 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7 frgrncvvdeq.v1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
87nbgrisvtx 28595 . . . . . . . 8 (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉)
98a1i 11 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
106, 9biimtrid 241 . . . . . 6 (𝜑 → (𝑥𝐷𝑥𝑉))
1110imp 407 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝑉)
12 frgrncvvdeq.y . . . . . 6 (𝜑𝑌𝑉)
1312adantr 481 . . . . 5 ((𝜑𝑥𝐷) → 𝑌𝑉)
14 frgrncvvdeq.xy . . . . . . 7 (𝜑𝑌𝐷)
15 elnelne2 3058 . . . . . . 7 ((𝑥𝐷𝑌𝐷) → 𝑥𝑌)
1614, 15sylan2 593 . . . . . 6 ((𝑥𝐷𝜑) → 𝑥𝑌)
1716ancoms 459 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝑌)
1811, 13, 173jca 1128 . . . 4 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
19 frgrncvvdeq.e . . . . 5 𝐸 = (Edg‘𝐺)
207, 19frcond3 29519 . . . 4 (𝐺 ∈ FriendGraph → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}))
214, 18, 20sylc 65 . . 3 ((𝜑𝑥𝐷) → ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛})
22 vex 3478 . . . . . . . . . 10 𝑛 ∈ V
23 elinsn 4714 . . . . . . . . . 10 ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
2422, 23mpan 688 . . . . . . . . 9 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25 frgrusgr 29511 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2619nbusgreledg 28607 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27 prcom 4736 . . . . . . . . . . . . . . . . . 18 {𝑛, 𝑥} = {𝑥, 𝑛}
2827eleq1i 2824 . . . . . . . . . . . . . . . . 17 ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
2926, 28bitrdi 286 . . . . . . . . . . . . . . . 16 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
3029biimpd 228 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
313, 25, 303syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
3231adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
3332com12 32 . . . . . . . . . . . 12 (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, 𝑛} ∈ 𝐸))
3433adantr 481 . . . . . . . . . . 11 ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → ((𝜑𝑥𝐷) → {𝑥, 𝑛} ∈ 𝐸))
3534imp 407 . . . . . . . . . 10 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → {𝑥, 𝑛} ∈ 𝐸)
361eqcomi 2741 . . . . . . . . . . . . . 14 (𝐺 NeighbVtx 𝑌) = 𝑁
3736eleq2i 2825 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛𝑁)
3837biimpi 215 . . . . . . . . . . . 12 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑁)
3938adantl 482 . . . . . . . . . . 11 ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛𝑁)
40 frgrncvvdeq.x . . . . . . . . . . . 12 (𝜑𝑋𝑉)
41 frgrncvvdeq.ne . . . . . . . . . . . 12 (𝜑𝑋𝑌)
42 frgrncvvdeq.a . . . . . . . . . . . 12 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
437, 19, 5, 1, 40, 12, 41, 14, 3, 42frgrncvvdeqlem2 29550 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
44 preq2 4738 . . . . . . . . . . . . 13 (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
4544eleq1d 2818 . . . . . . . . . . . 12 (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
4645riota2 7390 . . . . . . . . . . 11 ((𝑛𝑁 ∧ ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
4739, 43, 46syl2an 596 . . . . . . . . . 10 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
4835, 47mpbid 231 . . . . . . . . 9 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
4924, 48sylan 580 . . . . . . . 8 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
5049eqcomd 2738 . . . . . . 7 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → 𝑛 = (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
5150sneqd 4640 . . . . . 6 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
52 eqeq1 2736 . . . . . . 7 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5352adantr 481 . . . . . 6 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5451, 53mpbird 256 . . . . 5 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
5554ex 413 . . . 4 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5655rexlimivw 3151 . . 3 (∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
5721, 56mpcom 38 . 2 ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
582, 57eqtr2id 2785 1 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wnel 3046  wrex 3070  ∃!wreu 3374  Vcvv 3474  cin 3947  {csn 4628  {cpr 4630  cmpt 5231  cfv 6543  crio 7363  (class class class)co 7408  Vtxcvtx 28253  Edgcedg 28304  USGraphcusgr 28406   NeighbVtx cnbgr 28586   FriendGraph cfrgr 29508
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-2o 8466  df-oadd 8469  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-dju 9895  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-xnn0 12544  df-z 12558  df-uz 12822  df-fz 13484  df-hash 14290  df-edg 28305  df-upgr 28339  df-umgr 28340  df-usgr 28408  df-nbgr 28587  df-frgr 29509
This theorem is referenced by:  frgrncvvdeqlem5  29553
  Copyright terms: Public domain W3C validator