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Theorem opvtxfvi 28842
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 28837 . 2 ((𝑉 ∈ V ∧ 𝐞 ∈ V) → (Vtx‘⟚𝑉, 𝐞⟩) = 𝑉)
41, 2, 3mp2an 690 1 (Vtx‘⟚𝑉, 𝐞⟩) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3473  âŸšcop 4638  â€˜cfv 6553  Vtxcvtx 28829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-1st 7999  df-vtx 28831
This theorem is referenced by:  graop  28862  vtxvalsnop  28874  uhgrspanop  29129  fusgrfis  29163  cusgrsize  29288  fusgrmaxsize  29298  vtxdgop  29304  vtxdginducedm1  29377  vtxdginducedm1fi  29378  finsumvtxdg2ssteplem4  29382  finsumvtxdg2size  29384  eupth2lem3  30066  konigsberglem1  30082  konigsberglem2  30083  konigsberglem3  30084
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