Theorem frgrncvvdeq 27173

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 = (Vtx ` G )
frgrncvvdeq.d	|- D = (VtxDeg ` G )

Assertion

Ref	Expression
frgrncvvdeq	|- ( G e. FriendGraph -> A. x e. V A. y e. ( V \ { x } ) ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) )

Distinct variable groups:   x, G, y   x, V, y

Allowed substitution hints:   D( x, y)

Proof of Theorem frgrncvvdeq

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 6680	. . . . . 6 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( G NeighbVtx x ) e. _V )
2		frgrncvvdeq.v	. . . . . . 7 |- V = (Vtx ` G )
3		eqid 2622	. . . . . . 7 |- (Edg ` G ) = (Edg ` G )
4		eqid 2622	. . . . . . 7 |- ( G NeighbVtx x ) = ( G NeighbVtx x )
5		eqid 2622	. . . . . . 7 |- ( G NeighbVtx y ) = ( G NeighbVtx y )
6		simpl 473	. . . . . . . 8 |- ( ( x e. V /\ y e. ( V \ { x } ) ) -> x e. V )
7	6	ad2antlr 763	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> x e. V )
8		eldifi 3732	. . . . . . . . 9 |- ( y e. ( V \ { x } ) -> y e. V )
9	8	adantl 482	. . . . . . . 8 |- ( ( x e. V /\ y e. ( V \ { x } ) ) -> y e. V )
10	9	ad2antlr 763	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> y e. V )
11		eldif 3584	. . . . . . . . . 10 |- ( y e. ( V \ { x } ) <-> ( y e. V /\ -. y e. { x } ) )
12		velsn 4193	. . . . . . . . . . . . 13 |- ( y e. { x } <-> y = x )
13	12	biimpri 218	. . . . . . . . . . . 12 |- ( y = x -> y e. { x } )
14	13	equcoms 1947	. . . . . . . . . . 11 |- ( x = y -> y e. { x } )
15	14	necon3bi 2820	. . . . . . . . . 10 |- ( -. y e. { x } -> x =/= y )
16	11, 15	simplbiim 659	. . . . . . . . 9 |- ( y e. ( V \ { x } ) -> x =/= y )
17	16	adantl 482	. . . . . . . 8 |- ( ( x e. V /\ y e. ( V \ { x } ) ) -> x =/= y )
18	17	ad2antlr 763	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> x =/= y )
19		simpr 477	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> y e/ ( G NeighbVtx x ) )
20		simpl 473	. . . . . . . 8 |- ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) -> G e. FriendGraph )
21	20	adantr 481	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> G e. FriendGraph )
22		eqid 2622	. . . . . . 7 |- ( a e. ( G NeighbVtx x ) |-> ( iota_ b e. ( G NeighbVtx y ) { a , b } e. (Edg ` G ) ) ) = ( a e. ( G NeighbVtx x ) |-> ( iota_ b e. ( G NeighbVtx y ) { a , b } e. (Edg ` G ) ) )
23	2, 3, 4, 5, 7, 10, 18, 19, 21, 22	frgrncvvdeqlem10 27172	. . . . . 6 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( a e. ( G NeighbVtx x ) |-> ( iota_ b e. ( G NeighbVtx y ) { a , b } e. (Edg ` G ) ) ) : ( G NeighbVtx x ) -1-1-onto-> ( G NeighbVtx y ) )
24	1, 23	hasheqf1od 13144	. . . . 5 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( # ` ( G NeighbVtx x ) ) = ( # ` ( G NeighbVtx y ) ) )
25		frgrusgr 27124	. . . . . . . 8 |- ( G e. FriendGraph -> G e. USGraph )
26	25, 6	anim12i 590	. . . . . . 7 |- ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) -> ( G e. USGraph /\ x e. V ) )
27	26	adantr 481	. . . . . 6 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( G e. USGraph /\ x e. V ) )
28	2	hashnbusgrvd 26424	. . . . . 6 |- ( ( G e. USGraph /\ x e. V ) -> ( # ` ( G NeighbVtx x ) ) = ( (VtxDeg ` G ) ` x ) )
29	27, 28	syl 17	. . . . 5 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( # ` ( G NeighbVtx x ) ) = ( (VtxDeg ` G ) ` x ) )
30	25, 9	anim12i 590	. . . . . . 7 |- ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) -> ( G e. USGraph /\ y e. V ) )
31	30	adantr 481	. . . . . 6 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( G e. USGraph /\ y e. V ) )
32	2	hashnbusgrvd 26424	. . . . . 6 |- ( ( G e. USGraph /\ y e. V ) -> ( # ` ( G NeighbVtx y ) ) = ( (VtxDeg ` G ) ` y ) )
33	31, 32	syl 17	. . . . 5 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( # ` ( G NeighbVtx y ) ) = ( (VtxDeg ` G ) ` y ) )
34	24, 29, 33	3eqtr3d 2664	. . . 4 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( (VtxDeg ` G ) ` x ) = ( (VtxDeg ` G ) ` y ) )
35		frgrncvvdeq.d	. . . . 5 |- D = (VtxDeg ` G )
36	35	fveq1i 6192	. . . 4 |- ( D ` x ) = ( (VtxDeg ` G ) ` x )
37	35	fveq1i 6192	. . . 4 |- ( D ` y ) = ( (VtxDeg ` G ) ` y )
38	34, 36, 37	3eqtr4g 2681	. . 3 |- ( ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) /\ y e/ ( G NeighbVtx x ) ) -> ( D ` x ) = ( D ` y ) )
39	38	ex 450	. 2 |- ( ( G e. FriendGraph /\ ( x e. V /\ y e. ( V \ { x } ) ) ) -> ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) )
40	39	ralrimivva 2971	1 |- ( G e. FriendGraph -> A. x e. V A. y e. ( V \ { x } ) ( y e/ ( G NeighbVtx x ) -> ( D ` x ) = ( D ` y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   _Vcvv 3200   \ cdif 3571   {csn 4177   {cpr 4179   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224  VtxDegcvtxdg 26361   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-nbgr 26228  df-vtxdg 26362  df-frgr 27121

This theorem is referenced by:  frgrwopreglem4a  27174