Theorem grlimfn 47964

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 47963	. 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 6839	. . 3 ⊢ (Vtx‘ℎ) ∈ V
3		f1of 6768	. . . . 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 6841	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘𝑔) ∈ V)
6		id 22	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘ℎ) ∈ V)
7	4, 5, 6	fabexd 7877	. . 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 8012	1 ⊢ GraphLocIso Fn (V × V)

Syntax hints:   ∧ wa 395   ∈ wcel 2109  {cab 2707  ∀wral 3044  Vcvv 3438   class class class wbr 5095   × cxp 5621   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28959   ClNeighbVtx cclnbgr 47803   ISubGr cisubgr 47845   ≃_𝑔𝑟 cgric 47861   GraphLocIso cgrlim 47961

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-f1o 6493  df-fv 6494  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-grlim 47963

This theorem is referenced by:  brgrlic  47989  grlicrel  47991