Theorem grlimf1o 49288

Description: A local isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 21-May-2025.)

Hypotheses

Ref	Expression
grlimprop.v	⊢ 𝑉 = (Vtx‘𝐺)
grlimprop.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
grlimf1o	⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹:𝑉⟶1-1onto𝑊)

Proof of Theorem grlimf1o

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grlimprop.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		grlimprop.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐻)
3	1, 2	grlimprop 49287	. 2 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉⟶1-1onto𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))))
4	3	simpld 500	1 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹:𝑉⟶1-1onto𝑊)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5106  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443  Vtxcvtx 29604   ClNeighbVtx cclnbgr 49121   ISubGr cisubgr 49163   ≃_𝑔𝑟 cgric 49179   GraphLocIso cgrlim 49279

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-grlim 49281

This theorem is used by:  grlicen  49320