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Theorem opiedgfv 26784
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 5588 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opiedgval 26783 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
4 op2ndg 7694 . 2 ((𝑉𝑋𝐸𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
53, 4eqtrd 2854 1 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  Vcvv 3493  cop 4565   × cxp 5546  cfv 6348  2nd c2nd 7680  iEdgciedg 26774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-2nd 7682  df-iedg 26776
This theorem is referenced by:  opiedgov  26785  opiedgfvi  26787  gropd  26808  edgopval  26828  isuhgrop  26847  uhgrunop  26852  upgrop  26871  upgr0eop  26891  upgr1eop  26892  upgrunop  26896  umgrunop  26898  isuspgrop  26938  isusgrop  26939  ausgrusgrb  26942  usgr0eop  27020  uspgr1eop  27021  usgr1eop  27024  usgrexmpllem  27034  uhgrspan1lem3  27076  upgrres1lem3  27086  fusgrfisbase  27102  fusgrfisstep  27103  usgrexi  27215  cusgrexi  27217  p1evtxdeqlem  27286  p1evtxdeq  27287  p1evtxdp1  27288  uspgrloopiedg  27291  umgr2v2eiedg  27297  wlk2v2e  27928  eupthvdres  28006  eupth2lemb  28008  konigsbergiedg  28018  strisomgrop  43995  ushrisomgr  43996
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