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Theorem gricref 47827
Description: Graph isomorphism is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 29-Apr-2025.)
Assertion
Ref Expression
gricref (𝐺 ∈ UHGraph → 𝐺𝑔𝑟 𝐺)

Proof of Theorem gricref
StepHypRef Expression
1 grimid 47815 . 2 (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))
2 brgrici 47820 . 2 (( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺) → 𝐺𝑔𝑟 𝐺)
31, 2syl 17 1 (𝐺 ∈ UHGraph → 𝐺𝑔𝑟 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5148   I cid 5582  cres 5691  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  UHGraphcuhgr 29088   GraphIso cgrim 47799  𝑔𝑟 cgric 47800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-uhgr 29090  df-grim 47802  df-gric 47805
This theorem is referenced by:  gricer  47831  grlicref  47908
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