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Theorem opvtxfv 26105
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 5305 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opvtxval 26104 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1st ‘⟨𝑉, 𝐸⟩))
4 op1stg 7327 . 2 ((𝑉𝑋𝐸𝑌) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
53, 4eqtrd 2805 1 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   × cxp 5247  cfv 6031  1st c1st 7313  Vtxcvtx 26095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-vtx 26097
This theorem is referenced by:  opvtxov  26106  opvtxfvi  26110  graop  26142  gropd  26144  isuhgrop  26186  uhgrunop  26191  upgrop  26210  upgr1eop  26231  upgrunop  26235  umgrunop  26237  isuspgrop  26278  isusgrop  26279  ausgrusgrb  26282  uspgr1eop  26362  usgr1eop  26365  usgrexmpllem  26375  uhgrspanop  26411  uhgrspan1lem2  26416  upgrres1lem2  26426  opfusgr  26438  fusgrfisbase  26443  fusgrfisstep  26444  usgrexi  26572  cusgrexi  26574  fusgrmaxsize  26595  p1evtxdeqlem  26643  p1evtxdeq  26644  p1evtxdp1  26645  uspgrloopvtx  26646  umgr2v2evtx  26652  wlk2v2e  27337  eupthvdres  27415  eupth2lemb  27417  konigsbergvtx  27426  konigsberg  27437  uspgrsprfo  42284
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