Theorem grlimfn 47991

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 47990	. 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 6889	. . 3 ⊢ (Vtx‘ℎ) ∈ V
3		f1of 6818	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
4	3	ad2antrl 728	. . . 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 6891	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘𝑔) ∈ V)
6		id 22	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘ℎ) ∈ V)
7	4, 5, 6	fabexd 7933	. . 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 8069	1 ⊢ GraphLocIso Fn (V × V)

Syntax hints:   ∧ wa 395   ∈ wcel 2108  {cab 2713  ∀wral 3051  Vcvv 3459   class class class wbr 5119   × cxp 5652   Fn wfn 6526  ⟶wf 6527  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405  Vtxcvtx 28975   ClNeighbVtx cclnbgr 47832   ISubGr cisubgr 47873   ≃_𝑔𝑟 cgric 47889   GraphLocIso cgrlim 47988

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-f1o 6538  df-fv 6539  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-grlim 47990

This theorem is referenced by:  brgrlic  48009  grlicrel  48011