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Theorem graop 28269
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 6891 . . 3 (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3 fvex 6901 . . . 4 (Vtx‘𝐺) ∈ V
4 fvex 6901 . . . 4 (iEdg‘𝐺) ∈ V
53, 4opvtxfvi 28249 . . 3 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
62, 5eqtr2i 2762 . 2 (Vtx‘𝐺) = (Vtx‘𝐻)
71fveq2i 6891 . . 3 (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
83, 4opiedgfvi 28250 . . 3 (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
97, 8eqtr2i 2762 . 2 (iEdg‘𝐺) = (iEdg‘𝐻)
106, 9pm3.2i 472 1 ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  cop 4633  cfv 6540  Vtxcvtx 28236  iEdgciedg 28237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-1st 7970  df-2nd 7971  df-vtx 28238  df-iedg 28239
This theorem is referenced by: (None)
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