Theorem graop 29102

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 6837	. . 3 ⊢ (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3		fvex 6847	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
4		fvex 6847	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	3, 4	opvtxfvi 29082	. . 3 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
6	2, 5	eqtr2i 2760	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻)
7	1	fveq2i 6837	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	3, 4	opiedgfvi 29083	. . 3 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
9	7, 8	eqtr2i 2760	. 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 1541  ⟨cop 4586  ‘cfv 6492  Vtxcvtx 29069  iEdgciedg 29070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7933  df-2nd 7934  df-vtx 29071  df-iedg 29072

This theorem is referenced by: (None)