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Theorem grlimfn 47946
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 47945 . 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 6919 . . 3 (Vtx‘) ∈ V
3 f1of 6848 . . . . 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 6921 . . . 4 ((Vtx‘) ∈ V → (Vtx‘𝑔) ∈ V)
6 id 22 . . . 4 ((Vtx‘) ∈ V → (Vtx‘) ∈ V)
74, 5, 6fabexd 7959 . . 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 8095 1 GraphLocIso Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2108  {cab 2714  wral 3061  Vcvv 3480   class class class wbr 5143   × cxp 5683   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013   ClNeighbVtx cclnbgr 47805   ISubGr cisubgr 47846  𝑔𝑟 cgric 47862   GraphLocIso cgrlim 47943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grlim 47945
This theorem is referenced by:  brgrlic  47964  grlicrel  47966
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