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Theorem nbgraop 21441
 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 21438 . 2 Neighbors
2 opex 4430 . . . 4
32a1i 11 . . 3
4 op1stg 6362 . . . . . . . 8
54eqcomd 2443 . . . . . . 7
65eleq2d 2505 . . . . . 6
76biimpa 472 . . . . 5
87adantr 453 . . . 4
9 fveq2 5731 . . . . 5
109adantl 454 . . . 4
118, 10eleqtrrd 2515 . . 3
12 fvex 5745 . . . 4
13 rabexg 4356 . . . 4
1412, 13mp1i 12 . . 3
159, 4sylan9eq 2490 . . . . . . . . 9
1615ex 425 . . . . . . . 8
1716adantr 453 . . . . . . 7
1817com12 30 . . . . . 6
1918adantr 453 . . . . 5
2019imp 420 . . . 4
21 preq1 3885 . . . . . . 7
2221adantl 454 . . . . . 6
2322adantl 454 . . . . 5
24 fveq2 5731 . . . . . . . . . . . 12
25 op2ndg 6363 . . . . . . . . . . . 12
2624, 25sylan9eq 2490 . . . . . . . . . . 11
2726ex 425 . . . . . . . . . 10
2827adantr 453 . . . . . . . . 9
2928com12 30 . . . . . . . 8
3029adantr 453 . . . . . . 7
3130imp 420 . . . . . 6
3231rneqd 5100 . . . . 5
3323, 32eleq12d 2506 . . . 4
3420, 33rabeqbidv 2953 . . 3
353, 11, 14, 34ovmpt2dv2 6210 . 2 Neighbors Neighbors
361, 35mpi 17 1 Neighbors
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  crab 2711  cvv 2958  cpr 3817  cop 3819   crn 4882  cfv 5457  (class class class)co 6084   cmpt2 6086  c1st 6350  c2nd 6351   Neighbors cnbgra 21435 This theorem is referenced by:  nbgraop1  21442  nbgrael  21443  nbusgra  21445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-nbgra 21438
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