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Theorem rusgraprop2 24646
Description: The properties of a k-regular undirected simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
Assertion
Ref Expression
rusgraprop2  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  ( # `
 ( <. V ,  E >. Neighbors  p ) )  =  K ) )
Distinct variable groups:    E, p    K, p    V, p

Proof of Theorem rusgraprop2
StepHypRef Expression
1 rusgraprop 24633 . 2  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  (
( V VDeg  E ) `  p )  =  K ) )
2 simp1 996 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K )  ->  V USGrph  E )
3 simp2 997 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K )  ->  K  e.  NN0 )
4 hashnbgravdg 24617 . . . . . . 7  |-  ( ( V USGrph  E  /\  p  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  p ) )  =  ( ( V VDeg  E
) `  p )
)
54adantlr 714 . . . . . 6  |-  ( ( ( V USGrph  E  /\  K  e.  NN0 )  /\  p  e.  V )  ->  ( # `  ( <. V ,  E >. Neighbors  p
) )  =  ( ( V VDeg  E ) `
 p ) )
6 eqeq2 2482 . . . . . . 7  |-  ( K  =  ( ( V VDeg 
E ) `  p
)  ->  ( ( # `
 ( <. V ,  E >. Neighbors  p ) )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  p
) )  =  ( ( V VDeg  E ) `
 p ) ) )
76eqcoms 2479 . . . . . 6  |-  ( ( ( V VDeg  E ) `
 p )  =  K  ->  ( ( # `
 ( <. V ,  E >. Neighbors  p ) )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  p
) )  =  ( ( V VDeg  E ) `
 p ) ) )
85, 7syl5ibrcom 222 . . . . 5  |-  ( ( ( V USGrph  E  /\  K  e.  NN0 )  /\  p  e.  V )  ->  ( ( ( V VDeg 
E ) `  p
)  =  K  -> 
( # `  ( <. V ,  E >. Neighbors  p
) )  =  K ) )
98ralimdva 2872 . . . 4  |-  ( ( V USGrph  E  /\  K  e. 
NN0 )  ->  ( A. p  e.  V  ( ( V VDeg  E
) `  p )  =  K  ->  A. p  e.  V  ( # `  ( <. V ,  E >. Neighbors  p
) )  =  K ) )
1093impia 1193 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K )  ->  A. p  e.  V  ( # `  ( <. V ,  E >. Neighbors  p
) )  =  K )
112, 3, 103jca 1176 . 2  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K )  ->  ( V USGrph  E  /\  K  e.  NN0  /\ 
A. p  e.  V  ( # `  ( <. V ,  E >. Neighbors  p
) )  =  K ) )
121, 11syl 16 1  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  ( # `
 ( <. V ,  E >. Neighbors  p ) )  =  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   NN0cn0 10795   #chash 12373   USGrph cusg 24034   Neighbors cnbgra 24121   VDeg cvdg 24597   RegUSGrph crusgra 24627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-xadd 11319  df-fz 11673  df-hash 12374  df-usgra 24037  df-nbgra 24124  df-vdgr 24598  df-rgra 24628  df-rusgra 24629
This theorem is referenced by:  rusgraprop3  24647  numclwwlk1  24803
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