Theorem grimid 48385

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 2740	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) = (Vtx‘𝐺))
3		eqidd 2740	. 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
4	1, 1, 2, 3	grimidvtxedg 48384	1 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))

Syntax hints:   → wi 4   ∈ wcel 2119   I cid 5513   ↾ cres 5621  ‘cfv 6486  (class class class)co 7357  Vtxcvtx 29084  iEdgciedg 29085  UHGraphcuhgr 29144   GraphIso cgrim 48374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8766  df-uhgr 29146  df-grim 48377

This theorem is referenced by:  gricref  48419