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Theorem hashnbgravdg 25720
 Description: The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 25719. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
hashnbgravdg USGrph Neighbors VDeg

Proof of Theorem hashnbgravdg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgraf1o 25254 . . . 4 USGrph Neighbors
2 hasheqf1o 12570 . . . 4 Neighbors Neighbors Neighbors
31, 2syl5ibr 229 . . 3 Neighbors USGrph Neighbors
4 edgusgranbfin 25257 . . . . . 6 USGrph Neighbors
5 pm2.24 112 . . . . . 6 Neighbors
64, 5syl6bi 236 . . . . 5 USGrph Neighbors Neighbors
76com3l 83 . . . 4 Neighbors USGrph Neighbors
87imp 436 . . 3 Neighbors USGrph Neighbors
9 pm2.24 112 . . . . . 6 Neighbors Neighbors Neighbors
104, 9syl6bir 237 . . . . 5 USGrph Neighbors Neighbors
1110com13 82 . . . 4 Neighbors USGrph Neighbors
1211imp 436 . . 3 Neighbors USGrph Neighbors
13 ovex 6336 . . . . . . 7 Neighbors
1413a1i 11 . . . . . 6 USGrph Neighbors
15 simpl 464 . . . . . 6 Neighbors Neighbors
16 hashinf 12558 . . . . . 6 Neighbors Neighbors Neighbors
1714, 15, 16syl2anr 486 . . . . 5 Neighbors USGrph Neighbors
18 usgrav 25144 . . . . . . . . 9 USGrph
1918simprd 470 . . . . . . . 8 USGrph
20 dmexg 6743 . . . . . . . 8
21 rabexg 4549 . . . . . . . 8
2219, 20, 213syl 18 . . . . . . 7 USGrph
2322adantr 472 . . . . . 6 USGrph
24 simpr 468 . . . . . 6 Neighbors
25 hashinf 12558 . . . . . 6
2623, 24, 25syl2anr 486 . . . . 5 Neighbors USGrph
2717, 26eqtr4d 2508 . . . 4 Neighbors USGrph Neighbors
2827ex 441 . . 3 Neighbors USGrph Neighbors
293, 8, 12, 284cases 964 . 2 USGrph Neighbors
30 vdusgraval 25714 . 2 USGrph VDeg
3129, 30eqtr4d 2508 1 USGrph Neighbors VDeg
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 376   wceq 1452  wex 1671   wcel 1904  crab 2760  cvv 3031  cop 3965   class class class wbr 4395   cdm 4839  wf1o 5588  cfv 5589  (class class class)co 6308  cfn 7587   cpnf 9690  chash 12553   USGrph cusg 25136   Neighbors cnbgra 25224   VDeg cvdg 25700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-xadd 11433  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227  df-vdgr 25701 This theorem is referenced by:  nbhashnn0  25721  usgrauvtxvdbi  25727  cusgraisrusgra  25745  rusgraprop2  25749  frgrancvvdgeq  25850  usgreghash2spotv  25873  vdusgravaledg  40179  usgrauvtxvd  40180  vdcusgra  40181
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