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Theorem opvtxfvi 26721
Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
Hypotheses
Ref Expression
opvtxfvi.v 𝑉 ∈ V
opvtxfvi.e 𝐸 ∈ V
Assertion
Ref Expression
opvtxfvi (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Proof of Theorem opvtxfvi
StepHypRef Expression
1 opvtxfvi.v . 2 𝑉 ∈ V
2 opvtxfvi.e . 2 𝐸 ∈ V
3 opvtxfv 26716 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
41, 2, 3mp2an 688 1 (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  cop 4563  cfv 6348  Vtxcvtx 26708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  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-1st 7678  df-vtx 26710
This theorem is referenced by:  graop  26741  vtxvalsnop  26753  uhgrspanop  27005  fusgrfis  27039  cusgrsize  27163  fusgrmaxsize  27173  vtxdgop  27179  vtxdginducedm1  27252  vtxdginducedm1fi  27253  finsumvtxdg2ssteplem4  27257  finsumvtxdg2size  27259  eupth2lem3  27942  konigsberglem1  27958  konigsberglem2  27959  konigsberglem3  27960
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