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Theorem grlimfn 47882
Description: The graph local isomorphism function is a well-defined function. (Contributed by AV, 20-May-2025.)
Assertion
Ref Expression
grlimfn GraphLocIso Fn (V × V)

Proof of Theorem grlimfn
Dummy variables 𝑓 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-grlim 47881 . 2 GraphLocIso = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))})
2 fvex 6920 . . 3 (Vtx‘) ∈ V
3 f1of 6849 . . . . 5 (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘))
43ad2antrl 728 . . . 4 (((Vtx‘) ∈ V ∧ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘))
5 fvexd 6922 . . . 4 ((Vtx‘) ∈ V → (Vtx‘𝑔) ∈ V)
6 id 22 . . . 4 ((Vtx‘) ∈ V → (Vtx‘) ∈ V)
74, 5, 6fabexd 7958 . . 3 ((Vtx‘) ∈ V → {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))} ∈ V)
82, 7ax-mp 5 . 2 {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))} ∈ V
91, 8fnmpoi 8094 1 GraphLocIso Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  {cab 2712  wral 3059  Vcvv 3478   class class class wbr 5148   × cxp 5687   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   ClNeighbVtx cclnbgr 47743   ISubGr cisubgr 47784  𝑔𝑟 cgric 47800   GraphLocIso cgrlim 47879
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-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-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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-f1o 6570  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-grlim 47881
This theorem is referenced by:  brgrlic  47900  grlicrel  47902
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