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Theorem opvtxfv 29036
Description: The set of vertices 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
opvtxfv ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Proof of Theorem opvtxfv
StepHypRef Expression
1 opelvvg 5730 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opvtxval 29035 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
4 op1stg 8025 . 2 ((𝑉𝑋𝐸𝑌) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
53, 4eqtrd 2775 1 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   × cxp 5687  cfv 6563  1st c1st 8011  Vtxcvtx 29028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-vtx 29030
This theorem is referenced by:  opvtxov  29037  opvtxfvi  29041  gropd  29063  isuhgrop  29102  uhgrunop  29107  upgrop  29126  upgr1eop  29147  upgrunop  29151  umgrunop  29153  isuspgrop  29193  isusgrop  29194  ausgrusgrb  29197  uspgr1eop  29279  usgr1eop  29282  usgrexmpllem  29292  uhgrspan1lem2  29333  upgrres1lem2  29343  opfusgr  29355  fusgrfisbase  29360  fusgrfisstep  29361  usgrexi  29473  cusgrexi  29475  p1evtxdeqlem  29545  p1evtxdeq  29546  p1evtxdp1  29547  uspgrloopvtx  29548  umgr2v2evtx  29554  wlk2v2e  30186  eupthvdres  30264  eupth2lemb  30266  konigsbergvtx  30275  konigsberg  30286  isubgrvtx  47791  opstrgric  47833  ushggricedg  47834  usgrexmpl1vtx  47918  usgrexmpl2vtx  47923  uspgrsprfo  47992
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