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Theorem frgrncvvdeq 29829
Description: In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrncvvdeq.v 𝑉 = (Vtxβ€˜πΊ)
frgrncvvdeq.d 𝐷 = (VtxDegβ€˜πΊ)
Assertion
Ref Expression
frgrncvvdeq (𝐺 ∈ FriendGraph β†’ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ (𝑉 βˆ– {π‘₯})(𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)))
Distinct variable groups:   π‘₯,𝐺,𝑦   π‘₯,𝑉,𝑦
Allowed substitution hints:   𝐷(π‘₯,𝑦)

Proof of Theorem frgrncvvdeq
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7446 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (𝐺 NeighbVtx π‘₯) ∈ V)
2 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtxβ€˜πΊ)
3 eqid 2730 . . . . . . 7 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
4 eqid 2730 . . . . . . 7 (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx π‘₯)
5 eqid 2730 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6 simpl 481 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯})) β†’ π‘₯ ∈ 𝑉)
76ad2antlr 723 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ π‘₯ ∈ 𝑉)
8 eldifi 4125 . . . . . . . . 9 (𝑦 ∈ (𝑉 βˆ– {π‘₯}) β†’ 𝑦 ∈ 𝑉)
98adantl 480 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯})) β†’ 𝑦 ∈ 𝑉)
109ad2antlr 723 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ 𝑦 ∈ 𝑉)
11 eldif 3957 . . . . . . . . . 10 (𝑦 ∈ (𝑉 βˆ– {π‘₯}) ↔ (𝑦 ∈ 𝑉 ∧ Β¬ 𝑦 ∈ {π‘₯}))
12 velsn 4643 . . . . . . . . . . . . 13 (𝑦 ∈ {π‘₯} ↔ 𝑦 = π‘₯)
1312biimpri 227 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ 𝑦 ∈ {π‘₯})
1413equcoms 2021 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ 𝑦 ∈ {π‘₯})
1514necon3bi 2965 . . . . . . . . . 10 (Β¬ 𝑦 ∈ {π‘₯} β†’ π‘₯ β‰  𝑦)
1611, 15simplbiim 503 . . . . . . . . 9 (𝑦 ∈ (𝑉 βˆ– {π‘₯}) β†’ π‘₯ β‰  𝑦)
1716adantl 480 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯})) β†’ π‘₯ β‰  𝑦)
1817ad2antlr 723 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ π‘₯ β‰  𝑦)
19 simpr 483 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯))
20 simpl 481 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ 𝐺 ∈ FriendGraph )
2120adantr 479 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ 𝐺 ∈ FriendGraph )
22 eqid 2730 . . . . . . 7 (π‘Ž ∈ (𝐺 NeighbVtx π‘₯) ↦ (℩𝑏 ∈ (𝐺 NeighbVtx 𝑦){π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ))) = (π‘Ž ∈ (𝐺 NeighbVtx π‘₯) ↦ (℩𝑏 ∈ (𝐺 NeighbVtx 𝑦){π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ)))
232, 3, 4, 5, 7, 10, 18, 19, 21, 22frgrncvvdeqlem10 29828 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (π‘Ž ∈ (𝐺 NeighbVtx π‘₯) ↦ (℩𝑏 ∈ (𝐺 NeighbVtx 𝑦){π‘Ž, 𝑏} ∈ (Edgβ€˜πΊ))):(𝐺 NeighbVtx π‘₯)–1-1-ontoβ†’(𝐺 NeighbVtx 𝑦))
241, 23hasheqf1od 14317 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (β™―β€˜(𝐺 NeighbVtx π‘₯)) = (β™―β€˜(𝐺 NeighbVtx 𝑦)))
25 frgrusgr 29781 . . . . . . . 8 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
2625, 6anim12i 611 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ (𝐺 ∈ USGraph ∧ π‘₯ ∈ 𝑉))
2726adantr 479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (𝐺 ∈ USGraph ∧ π‘₯ ∈ 𝑉))
282hashnbusgrvd 29052 . . . . . 6 ((𝐺 ∈ USGraph ∧ π‘₯ ∈ 𝑉) β†’ (β™―β€˜(𝐺 NeighbVtx π‘₯)) = ((VtxDegβ€˜πΊ)β€˜π‘₯))
2927, 28syl 17 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (β™―β€˜(𝐺 NeighbVtx π‘₯)) = ((VtxDegβ€˜πΊ)β€˜π‘₯))
3025, 9anim12i 611 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
3130adantr 479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
322hashnbusgrvd 29052 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑦)) = ((VtxDegβ€˜πΊ)β€˜π‘¦))
3331, 32syl 17 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑦)) = ((VtxDegβ€˜πΊ)β€˜π‘¦))
3424, 29, 333eqtr3d 2778 . . . 4 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ ((VtxDegβ€˜πΊ)β€˜π‘₯) = ((VtxDegβ€˜πΊ)β€˜π‘¦))
35 frgrncvvdeq.d . . . . 5 𝐷 = (VtxDegβ€˜πΊ)
3635fveq1i 6891 . . . 4 (π·β€˜π‘₯) = ((VtxDegβ€˜πΊ)β€˜π‘₯)
3735fveq1i 6891 . . . 4 (π·β€˜π‘¦) = ((VtxDegβ€˜πΊ)β€˜π‘¦)
3834, 36, 373eqtr4g 2795 . . 3 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦))
3938ex 411 . 2 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)))
4039ralrimivva 3198 1 (𝐺 ∈ FriendGraph β†’ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ (𝑉 βˆ– {π‘₯})(𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   βˆ‰ wnel 3044  βˆ€wral 3059  Vcvv 3472   βˆ– cdif 3944  {csn 4627  {cpr 4629   ↦ cmpt 5230  β€˜cfv 6542  β„©crio 7366  (class class class)co 7411  β™―chash 14294  Vtxcvtx 28523  Edgcedg 28574  USGraphcusgr 28676   NeighbVtx cnbgr 28856  VtxDegcvtxdg 28989   FriendGraph cfrgr 29778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-xadd 13097  df-fz 13489  df-hash 14295  df-edg 28575  df-uhgr 28585  df-ushgr 28586  df-upgr 28609  df-umgr 28610  df-uspgr 28677  df-usgr 28678  df-nbgr 28857  df-vtxdg 28990  df-frgr 29779
This theorem is referenced by:  frgrwopreglem4a  29830
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