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Theorem graop 26741
Description: Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
Hypothesis
Ref Expression
graop.h 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
Assertion
Ref Expression
graop ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Proof of Theorem graop
StepHypRef Expression
1 graop.h . . . 4 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
21fveq2i 6666 . . 3 (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3 fvex 6676 . . . 4 (Vtx‘𝐺) ∈ V
4 fvex 6676 . . . 4 (iEdg‘𝐺) ∈ V
53, 4opvtxfvi 26721 . . 3 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
62, 5eqtr2i 2842 . 2 (Vtx‘𝐺) = (Vtx‘𝐻)
71fveq2i 6666 . . 3 (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
83, 4opiedgfvi 26722 . . 3 (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
97, 8eqtr2i 2842 . 2 (iEdg‘𝐺) = (iEdg‘𝐻)
106, 9pm3.2i 471 1 ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  cop 4563  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709
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-2nd 7679  df-vtx 26710  df-iedg 26711
This theorem is referenced by: (None)
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