Theorem graop 29047

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

Step	Hyp	Ref	Expression
1		graop.h	. . . 4 ⊢ 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
2	1	fveq2i 6908	. . 3 ⊢ (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3		fvex 6918	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
4		fvex 6918	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	3, 4	opvtxfvi 29027	. . 3 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
6	2, 5	eqtr2i 2765	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻)
7	1	fveq2i 6908	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	3, 4	opiedgfvi 29028	. . 3 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
9	7, 8	eqtr2i 2765	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘𝐻)
10	6, 9	pm3.2i 470	1 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Syntax hints:   ∧ wa 395   = wceq 1539  ⟨cop 4631  ‘cfv 6560  Vtxcvtx 29014  iEdgciedg 29015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fv 6568  df-1st 8015  df-2nd 8016  df-vtx 29016  df-iedg 29017

This theorem is referenced by: (None)