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Theorem dfnbgr3 39418
 Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves [see also nbgraop1 25165]. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.)
Hypotheses
Ref Expression
dfnbgr3.v Vtx
dfnbgr3.i iEdg
Assertion
Ref Expression
dfnbgr3 NeighbVtx
Distinct variable groups:   ,   ,,   ,,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem dfnbgr3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfnbgr3.v . . . 4 Vtx
2 eqid 2453 . . . 4 Edg Edg
31, 2nbgrval 39416 . . 3 NeighbVtx Edg
433adant3 1029 . 2 NeighbVtx Edg
5 edgaval 39222 . . . . . . 7 Edg iEdg
6 dfnbgr3.i . . . . . . . . 9 iEdg
76eqcomi 2462 . . . . . . . 8 iEdg
87rneqi 5064 . . . . . . 7 iEdg
95, 8syl6eq 2503 . . . . . 6 Edg
109rexeqdv 2996 . . . . 5 Edg
11103ad2ant1 1030 . . . 4 Edg
12 funfn 5614 . . . . . . 7
1312biimpi 198 . . . . . 6
14133ad2ant3 1032 . . . . 5
15 sseq2 3456 . . . . . 6
1615rexrn 6029 . . . . 5
1714, 16syl 17 . . . 4
1811, 17bitrd 257 . . 3 Edg
1918rabbidv 3038 . 2 Edg
204, 19eqtrd 2487 1 NeighbVtx
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   w3a 986   wceq 1446   wcel 1889  wrex 2740  crab 2743   cdif 3403   wss 3406  csn 3970  cpr 3972   cdm 4837   crn 4838   wfun 5579   wfn 5580  cfv 5585  (class class class)co 6295  Vtxcvtx 39111  iEdgciedg 39112  Edgcedga 39220   NeighbVtx cnbgr 39407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-edga 39221  df-nbgr 39411 This theorem is referenced by: (None)
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