Theorem grimid 47917

Description: The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and itself. (Contributed by AV, 29-Apr-2025.) (Prove shortened by AV, 5-May-2025.)

Assertion

Ref	Expression
grimid	⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))

Proof of Theorem grimid

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
2		eqidd 2732	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) = (Vtx‘𝐺))
3		eqidd 2732	. 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
4	1, 1, 2, 3	grimidvtxedg 47916	1 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))

Syntax hints:   → wi 4   ∈ wcel 2111   I cid 5505   ↾ cres 5613  ‘cfv 6476  (class class class)co 7341  Vtxcvtx 28969  iEdgciedg 28970  UHGraphcuhgr 29029   GraphIso cgrim 47906

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-map 8747  df-uhgr 29031  df-grim 47909

This theorem is referenced by:  gricref  47951