Theorem graop 29320

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 6885	. . 3 ⊢ (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3		fvex 6895	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
4		fvex 6895	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	3, 4	opvtxfvi 29300	. . 3 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
6	2, 5	eqtr2i 2793	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻)
7	1	fveq2i 6885	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	3, 4	opiedgfvi 29301	. . 3 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
9	7, 8	eqtr2i 2793	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘𝐻)
10	6, 9	pm3.2i 475	1 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Syntax hints:   ∧ wa 400   = wceq 1567  ⟨cop 4600  ‘cfv 6537  Vtxcvtx 29287  iEdgciedg 29288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7986  df-2nd 7987  df-vtx 29289  df-iedg 29290

This theorem is referenced by: (None)