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Theorem nbgraop 25230
 Description: The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
Assertion
Ref Expression
nbgraop Neighbors
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem nbgraop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 25227 . 2 Neighbors
2 opex 4664 . . . 4
32a1i 11 . . 3
4 op1stg 6824 . . . . . . . 8
54eqcomd 2477 . . . . . . 7
65eleq2d 2534 . . . . . 6
76biimpa 492 . . . . 5
87adantr 472 . . . 4
9 fveq2 5879 . . . . 5
109adantl 473 . . . 4
118, 10eleqtrrd 2552 . . 3
12 fvex 5889 . . . 4
13 rabexg 4549 . . . 4
1412, 13mp1i 13 . . 3
159, 4sylan9eq 2525 . . . . . . . . 9
1615ex 441 . . . . . . . 8
1716adantr 472 . . . . . . 7
1817com12 31 . . . . . 6
1918adantr 472 . . . . 5
2019imp 436 . . . 4
21 preq1 4042 . . . . . . 7
2221adantl 473 . . . . . 6
2322adantl 473 . . . . 5
24 fveq2 5879 . . . . . . . . . . . 12
25 op2ndg 6825 . . . . . . . . . . . 12
2624, 25sylan9eq 2525 . . . . . . . . . . 11
2726ex 441 . . . . . . . . . 10
2827adantr 472 . . . . . . . . 9
2928com12 31 . . . . . . . 8
3029adantr 472 . . . . . . 7
3130imp 436 . . . . . 6
3231rneqd 5068 . . . . 5
3323, 32eleq12d 2543 . . . 4
3420, 33rabeqbidv 3026 . . 3
353, 11, 14, 34ovmpt2dv2 6449 . 2 Neighbors Neighbors
361, 35mpi 20 1 Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  crab 2760  cvv 3031  cpr 3961  cop 3965   crn 4840  cfv 5589  (class class class)co 6308   cmpt2 6310  c1st 6810  c2nd 6811   Neighbors cnbgra 25224 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  ax-un 6602 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-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-nbgra 25227 This theorem is referenced by:  nbgraop1  25232  nbgrael  25233  nbusgra  25235  rusgraprop3  25750
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