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Theorem nbgrval 39570
 Description: The set of neighbors of a vertex in a graph . (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
nbgrval.v Vtx
nbgrval.e Edg
Assertion
Ref Expression
nbgrval NeighbVtx
Distinct variable groups:   ,   ,,   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem nbgrval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgr 39565 . 2 NeighbVtx Vtx Vtx Edg
2 nbgrval.v . . . 4 Vtx
321vgrex 39257 . . 3
4 fveq2 5879 . . . . . 6 Vtx Vtx
54, 2syl6reqr 2524 . . . . 5 Vtx
65eleq2d 2534 . . . 4 Vtx
76biimpac 494 . . 3 Vtx
8 fvex 5889 . . . . 5 Vtx
98difexi 4546 . . . 4 Vtx
10 rabexg 4549 . . . 4 Vtx Vtx Edg
119, 10mp1i 13 . . 3 Vtx Edg
124, 2syl6eqr 2523 . . . . . . 7 Vtx
1312adantr 472 . . . . . 6 Vtx
14 sneq 3969 . . . . . . 7
1514adantl 473 . . . . . 6
1613, 15difeq12d 3541 . . . . 5 Vtx
1716adantl 473 . . . 4 Vtx
18 fveq2 5879 . . . . . . . 8 Edg Edg
19 nbgrval.e . . . . . . . 8 Edg
2018, 19syl6eqr 2523 . . . . . . 7 Edg
2120adantr 472 . . . . . 6 Edg
2221adantl 473 . . . . 5 Edg
23 preq1 4042 . . . . . . . 8
2423sseq1d 3445 . . . . . . 7
2524adantl 473 . . . . . 6
2625adantl 473 . . . . 5
2722, 26rexeqbidv 2988 . . . 4 Edg
2817, 27rabeqbidv 3026 . . 3 Vtx Edg
293, 7, 11, 28ovmpt2dv2 6449 . 2 NeighbVtx Vtx Vtx Edg NeighbVtx
301, 29mpi 20 1 NeighbVtx
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904  wrex 2757  crab 2760  cvv 3031   cdif 3387   wss 3390  csn 3959  cpr 3961  cfv 5589  (class class class)co 6308   cmpt2 6310  Vtxcvtx 39251  Edgcedga 39371   NeighbVtx cnbgr 39561 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-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-nbgr 39565 This theorem is referenced by:  dfnbgr2  39571  dfnbgr3  39572  nbgrel  39574  nbuhgr  39575  nbupgr  39576  nbumgrvtx  39578  nbgr0vtxlem  39587  nbgrnself  39593
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