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Theorem graop 26820
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 6655 . . 3 (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3 fvex 6665 . . . 4 (Vtx‘𝐺) ∈ V
4 fvex 6665 . . . 4 (iEdg‘𝐺) ∈ V
53, 4opvtxfvi 26800 . . 3 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
62, 5eqtr2i 2846 . 2 (Vtx‘𝐺) = (Vtx‘𝐻)
71fveq2i 6655 . . 3 (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
83, 4opiedgfvi 26801 . . 3 (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
97, 8eqtr2i 2846 . 2 (iEdg‘𝐺) = (iEdg‘𝐻)
106, 9pm3.2i 474 1 ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  cop 4545  cfv 6334  Vtxcvtx 26787  iEdgciedg 26788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fv 6342  df-1st 7675  df-2nd 7676  df-vtx 26789  df-iedg 26790
This theorem is referenced by: (None)
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