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Theorem opiedgfv 25821
Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
opiedgfv ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Proof of Theorem opiedgfv
StepHypRef Expression
1 opelvvg 5135 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opiedgval 25820 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
4 op2ndg 7141 . 2 ((𝑉𝑋𝐸𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
53, 4eqtrd 2655 1 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cop 4161   × cxp 5082  cfv 5857  2nd c2nd 7127  iEdgciedg 25809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-2nd 7129  df-iedg 25811
This theorem is referenced by:  opiedgov  25822  opiedgfvi  25824  graop  25855  gropd  25857  edgopval  25876  isuhgrop  25895  uhgrunop  25900  upgr0eop  25938  upgr1eop  25939  upgrunop  25943  umgrunop  25945  isuspgrop  25983  isusgrop  25984  ausgrusgrb  25987  usgr0eop  26065  uspgr1eop  26066  usgr1eop  26069  usgrexmpllem  26079  griedg0ssusgr  26084  uhgrspanop  26115  uhgrspan1lem3  26121  upgrres1lem3  26126  fusgrfisbase  26142  fusgrfisstep  26143  usgrexi  26258  cusgrexi  26260  p1evtxdeqlem  26328  p1evtxdeq  26329  p1evtxdp1  26330  uspgrloopiedg  26333  umgr2v2eiedg  26339  rgrusgrprc  26389  wlk2v2e  26917  eupthvdres  26995  eupth2lemb  26997  konigsbergiedg  27007
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