Theorem grlimfn 48785

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 48784	. 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 6901	. . 3 ⊢ (Vtx‘ℎ) ∈ V
3		f1of 6827	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
4	3	ad2antrl 741	. . . 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 6903	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘𝑔) ∈ V)
6		id 23	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘ℎ) ∈ V)
7	4, 5, 6	fabexd 7943	. . 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 8076	1 ⊢ GraphLocIso Fn (V × V)

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2146  {cab 2744  ∀wral 3082  Vcvv 3458   class class class wbr 5114   × cxp 5664   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Vtxcvtx 29383   ClNeighbVtx cclnbgr 48624   ISubGr cisubgr 48666   ≃_𝑔𝑟 cgric 48682   GraphLocIso cgrlim 48782

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-f1o 6550  df-fv 6551  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-grlim 48784

This theorem is used by:  brgrlic  48810  grlicrel  48812