Theorem dfgrlic3 48006

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 48000	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2		n0 4319	. . 3 ⊢ ((𝐺 GraphLocIso 𝐻) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻))
3	1, 2	bitri 275	. 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 47986	. . . 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 1921	. 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 283	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)))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   class class class wbr 5110  dom cdm 5641   “ cima 5644  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  iEdgciedg 28931   ClNeighbVtx cclnbgr 47823   GraphLocIso cgrlim 47979   ≃_𝑙𝑔𝑟 cgrlic 47980

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-map 8804  df-vtx 28932  df-iedg 28933  df-clnbgr 47824  df-isubgr 47865  df-grim 47882  df-gric 47885  df-grlim 47981  df-grlic 47984

This theorem is referenced by:  grilcbri2  48007