Theorem graop 26822

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 6648	. . 3 ⊢ (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
3		fvex 6658	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
4		fvex 6658	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	3, 4	opvtxfvi 26802	. . 3 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺)
6	2, 5	eqtr2i 2822	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻)
7	1	fveq2i 6648	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
8	3, 4	opiedgfvi 26803	. . 3 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺)
9	7, 8	eqtr2i 2822	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘𝐻)
10	6, 9	pm3.2i 474	1 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Syntax hints:   ∧ wa 399   = wceq 1538  ⟨cop 4531  ‘cfv 6324  Vtxcvtx 26789  iEdgciedg 26790

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-1st 7671  df-2nd 7672  df-vtx 26791  df-iedg 26792

This theorem is referenced by: (None)