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Theorem opvtxfv 25681
 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 5087 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opvtxval 25680 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
4 op1stg 7071 . 2 ((𝑉𝑋𝐸𝑌) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
53, 4eqtrd 2644 1 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036  ‘cfv 5804  1st c1st 7057  Vtxcvtx 25673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-vtx 25675 This theorem is referenced by:  opvtxov  25682  opvtxfvi  25686  graop  25706  gropd  25708  isuhgrop  25736  uhgrunop  25741  upgr1eop  25781  upgrunop  25785  umgrunop  25787  isusgrop  40392  ausgrusgrb  40395  uspgr1eop  40473  usgr1eop  40476  usgrexmpllem  40484  uhgrspanop  40520  uhgrspan1lem2  40525  upgrres1lem2  40530  opfusgr  40542  fusgrfisbase  40547  fusgrfisstep  40548  usgrexi  40661  cusgrexi  40662  cusgrsize  40670  fusgrmaxsize  40680  p1evtxdeqlem  40728  p1evtxdeq  40729  p1evtxdp1  40730  uspgrloopvtx  40731  umgr2v2evtx  40737  1wlk2v2e  41324  eupthvdres  41403  eupth2lemb  41405  konigsbergvtx  41414  konigsberg-av  41427
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