Theorem dfgrlic3 49030

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)
dfgrlic3.i	⊢ 𝐼 = (iEdg‘𝐺)
dfgrlic3.j	⊢ 𝐽 = (iEdg‘𝐻)
dfgrlic3.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
dfgrlic3.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣))
dfgrlic3.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
dfgrlic3.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

Assertion

Ref	Expression
dfgrlic3	⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))

Distinct variable groups:   𝑓,𝐺,𝑣   𝑓,𝐻,𝑣   𝑣,𝑉   𝑓,𝑋   𝑓,𝑌   𝑔,𝐺,𝑖,𝑗,𝑓,𝑣   𝑥,𝐺   𝑔,𝐻,𝑖,𝑗   𝑥,𝐻   𝑖,𝐼,𝑥   𝑖,𝐽,𝑥   𝑖,𝐾   𝑔,𝑋,𝑗,𝑣   𝑖,𝐿   𝑔,𝑀,𝑖,𝑗   𝑥,𝑀   𝑔,𝑁,𝑖,𝑗   𝑥,𝑁   𝑖,𝑋   𝑖,𝑌,𝑗,𝑔,𝑣

Allowed substitution hints:   𝐼(𝑣, 𝑓, 𝑔, 𝑗)   𝐽(𝑣, 𝑓, 𝑔, 𝑗)   𝐾(𝑥, 𝑣, 𝑓, 𝑔, 𝑗)   𝐿(𝑥, 𝑣, 𝑓, 𝑔, 𝑗)   𝑀(𝑣, 𝑓)   𝑁(𝑣, 𝑓)   𝑉(𝑥, 𝑓, 𝑔, 𝑖, 𝑗)   𝑊(𝑥, 𝑣, 𝑓, 𝑔, 𝑖, 𝑗)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem dfgrlic3

Step	Hyp	Ref	Expression
1		brgrlic 49024	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2		n0 4299	. . 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		dfgrlic3.n	. . . . 5 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
7		dfgrlic3.m	. . . . 5 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣))
8		dfgrlic3.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
9		dfgrlic3.j	. . . . 5 ⊢ 𝐽 = (iEdg‘𝐻)
10		dfgrlic3.k	. . . . 5 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
11		dfgrlic3.l	. . . . 5 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
12	4, 5, 6, 7, 8, 9, 10, 11	isgrlim2 49003	. . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝑓 ∈ V) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
13	12	el3v3 3459	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
14	13	exbidv 1954	. 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
15	3, 14	bitrid 286	1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3898  ∅c0 4278   class class class wbr 5102  dom cdm 5647   “ cima 5650  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408  Vtxcvtx 29507  iEdgciedg 29508   ClNeighbVtx cclnbgr 48838   GraphLocIso cgrlim 48996   ≃_𝑙𝑔𝑟 cgrlic 48997

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 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-1o 8454  df-map 8827  df-vtx 29509  df-iedg 29510  df-clnbgr 48839  df-isubgr 48881  df-grim 48898  df-gric 48901  df-grlim 48998  df-grlic 49001

This theorem is used by:  grilcbri2  49031