Theorem dfgrlic2 48924

Description: Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)

Hypotheses

Ref	Expression
dfgrlic2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfgrlic2.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
dfgrlic2	⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))

Distinct variable groups:   𝑓,𝐺,𝑣   𝑓,𝐻,𝑣   𝑣,𝑉   𝑓,𝑋   𝑓,𝑌

Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑣, 𝑓)   𝑋(𝑣)   𝑌(𝑣)

Proof of Theorem dfgrlic2

Step	Hyp	Ref	Expression
1		brgrlic 48920	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2		n0 4300	. . 3 ⊢ ((𝐺 GraphLocIso 𝐻) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻))
3	1, 2	bitri 278	. 2 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻))
4		dfgrlic2.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5		dfgrlic2.w	. . . . 5 ⊢ 𝑊 = (Vtx‘𝐻)
6	4, 5	isgrlim 48898	. . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝑓 ∈ V) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))
7	6	el3v3 3459	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))
8	7	exbidv 1954	. 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))
9	3, 8	bitrid 286	1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4279   class class class wbr 5103  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7413  Vtxcvtx 29453   ClNeighbVtx cclnbgr 48734   ISubGr cisubgr 48776   ≃_𝑔𝑟 cgric 48792   GraphLocIso cgrlim 48892   ≃_𝑙𝑔𝑟 cgrlic 48893

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-1o 8455  df-grlim 48894  df-grlic 48897

This theorem is used by:  grilcbri  48925  grlicref  48928  grlicsym  48929  grlictr  48931  clnbgr3stgrgrlic  48936