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Theorem frgrncvvdeq 29295
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 7397 . . . . . 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 2737 . . . . . . 7 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
4 eqid 2737 . . . . . . 7 (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx π‘₯)
5 eqid 2737 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6 simpl 484 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯})) β†’ π‘₯ ∈ 𝑉)
76ad2antlr 726 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ π‘₯ ∈ 𝑉)
8 eldifi 4091 . . . . . . . . 9 (𝑦 ∈ (𝑉 βˆ– {π‘₯}) β†’ 𝑦 ∈ 𝑉)
98adantl 483 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯})) β†’ 𝑦 ∈ 𝑉)
109ad2antlr 726 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ 𝑦 ∈ 𝑉)
11 eldif 3925 . . . . . . . . . 10 (𝑦 ∈ (𝑉 βˆ– {π‘₯}) ↔ (𝑦 ∈ 𝑉 ∧ Β¬ 𝑦 ∈ {π‘₯}))
12 velsn 4607 . . . . . . . . . . . . 13 (𝑦 ∈ {π‘₯} ↔ 𝑦 = π‘₯)
1312biimpri 227 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ 𝑦 ∈ {π‘₯})
1413equcoms 2024 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ 𝑦 ∈ {π‘₯})
1514necon3bi 2971 . . . . . . . . . 10 (Β¬ 𝑦 ∈ {π‘₯} β†’ π‘₯ β‰  𝑦)
1611, 15simplbiim 506 . . . . . . . . 9 (𝑦 ∈ (𝑉 βˆ– {π‘₯}) β†’ π‘₯ β‰  𝑦)
1716adantl 483 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯})) β†’ π‘₯ β‰  𝑦)
1817ad2antlr 726 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ π‘₯ β‰  𝑦)
19 simpr 486 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯))
20 simpl 484 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ 𝐺 ∈ FriendGraph )
2120adantr 482 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ 𝐺 ∈ FriendGraph )
22 eqid 2737 . . . . . . 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 29294 . . . . . 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 14260 . . . . 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 29247 . . . . . . . 8 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
2625, 6anim12i 614 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ (𝐺 ∈ USGraph ∧ π‘₯ ∈ 𝑉))
2726adantr 482 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (𝐺 ∈ USGraph ∧ π‘₯ ∈ 𝑉))
282hashnbusgrvd 28518 . . . . . 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 614 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) β†’ (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
3130adantr 482 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
322hashnbusgrvd 28518 . . . . . 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 2785 . . . 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 6848 . . . 4 (π·β€˜π‘₯) = ((VtxDegβ€˜πΊ)β€˜π‘₯)
3735fveq1i 6848 . . . 4 (π·β€˜π‘¦) = ((VtxDegβ€˜πΊ)β€˜π‘¦)
3834, 36, 373eqtr4g 2802 . . 3 (((𝐺 ∈ FriendGraph ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 βˆ– {π‘₯}))) ∧ 𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯)) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦))
3938ex 414 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ‰ wnel 3050  βˆ€wral 3065  Vcvv 3448   βˆ– cdif 3912  {csn 4591  {cpr 4593   ↦ cmpt 5193  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362  β™―chash 14237  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142   NeighbVtx cnbgr 28322  VtxDegcvtxdg 28455   FriendGraph cfrgr 29244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-xadd 13041  df-fz 13432  df-hash 14238  df-edg 28041  df-uhgr 28051  df-ushgr 28052  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-nbgr 28323  df-vtxdg 28456  df-frgr 29245
This theorem is referenced by:  frgrwopreglem4a  29296
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