Theorem grlimfn 47803

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 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-grlim 47802	. 2 ⊢ GraphLocIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))})
2		fvex 6933	. . 3 ⊢ (Vtx‘ℎ) ∈ V
3		f1of 6862	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
4	3	ad2antrl 727	. . . 4 ⊢ (((Vtx‘ℎ) ∈ V ∧ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
5		fvexd 6935	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘𝑔) ∈ V)
6		id 22	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘ℎ) ∈ V)
7	4, 5, 6	fabexd 7975	. . 3 ⊢ ((Vtx‘ℎ) ∈ V → {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))} ∈ V)
8	2, 7	ax-mp 5	. 2 ⊢ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))} ∈ V
9	1, 8	fnmpoi 8111	1 ⊢ GraphLocIso Fn (V × V)

Syntax hints:   ∧ wa 395   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488   class class class wbr 5166   × cxp 5698   Fn wfn 6568  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031   ClNeighbVtx cclnbgr 47692   ISubGr cisubgr 47732   ≃_𝑔𝑟 cgric 47746   GraphLocIso cgrlim 47800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-f1o 6580  df-fv 6581  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-grlim 47802

This theorem is referenced by:  brgrlic  47821  grlicrel  47823