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Theorem graop 29064
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 6923 . . 3 (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3 fvex 6933 . . . 4 (Vtx‘𝐺) ∈ V
4 fvex 6933 . . . 4 (iEdg‘𝐺) ∈ V
53, 4opvtxfvi 29044 . . 3 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
62, 5eqtr2i 2769 . 2 (Vtx‘𝐺) = (Vtx‘𝐻)
71fveq2i 6923 . . 3 (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
83, 4opiedgfvi 29045 . . 3 (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
97, 8eqtr2i 2769 . 2 (iEdg‘𝐺) = (iEdg‘𝐻)
106, 9pm3.2i 470 1 ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  cop 4654  cfv 6573  Vtxcvtx 29031  iEdgciedg 29032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-1st 8030  df-2nd 8031  df-vtx 29033  df-iedg 29034
This theorem is referenced by: (None)
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