Theorem grlimfn 48333

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 48332	. 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 6855	. . 3 ⊢ (Vtx‘ℎ) ∈ V
3		f1of 6782	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
4	3	ad2antrl 729	. . . 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 6857	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘𝑔) ∈ V)
6		id 22	. . . 4 ⊢ ((Vtx‘ℎ) ∈ V → (Vtx‘ℎ) ∈ V)
7	4, 5, 6	fabexd 7889	. . 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 8024	1 ⊢ GraphLocIso Fn (V × V)

Syntax hints:   ∧ wa 395   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3442   class class class wbr 5100   × cxp 5630   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  Vtxcvtx 29081   ClNeighbVtx cclnbgr 48172   ISubGr cisubgr 48214   ≃_𝑔𝑟 cgric 48230   GraphLocIso cgrlim 48330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-f1o 6507  df-fv 6508  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-grlim 48332

This theorem is referenced by:  brgrlic  48358  grlicrel  48360