Theorem isgrlim2 47981

Description: A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025.)

Hypotheses

Ref	Expression
isgrlim.v	⊢ 𝑉 = (Vtx‘𝐺)
isgrlim.w	⊢ 𝑊 = (Vtx‘𝐻)
isgrlim2.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
isgrlim2.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
isgrlim2.i	⊢ 𝐼 = (iEdg‘𝐺)
isgrlim2.j	⊢ 𝐽 = (iEdg‘𝐻)
isgrlim2.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
isgrlim2.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

Assertion

Ref	Expression
isgrlim2	⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))

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

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

Proof of Theorem isgrlim2

Step	Hyp	Ref	Expression
1		isgrlim.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		isgrlim.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐻)
3	1, 2	isgrlim 47980	. 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))))
4		isgrlim2.n	. . . . . . . . 9 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
5	4	eqcomi 2738	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) = 𝑁
6	5	oveq2i 7351	. . . . . . 7 ⊢ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr 𝑁)
7		isgrlim2.m	. . . . . . . . 9 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
8	7	eqcomi 2738	. . . . . . . 8 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝑣)) = 𝑀
9	8	oveq2i 7351	. . . . . . 7 ⊢ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) = (𝐻 ISubGr 𝑀)
10	6, 9	breq12i 5097	. . . . . 6 ⊢ ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) ↔ (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀))
11	10	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) ↔ (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀)))
12		isgrlim2.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
13		isgrlim2.j	. . . . . . 7 ⊢ 𝐽 = (iEdg‘𝐻)
14		isgrlim2.k	. . . . . . 7 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
15		isgrlim2.l	. . . . . . 7 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
16	12, 13, 4, 7, 14, 15	clnbgrisubgrgrim 47930	. . . . . 6 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
17	16	3adant3 1132	. . . . 5 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
18	11, 17	bitrd 279	. . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
19	18	ralbidv 3152	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) ↔ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
20	19	anbi2d 630	. 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → ((𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
21	3, 20	bitrd 279	1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  {crab 3392   ⊆ wss 3899   class class class wbr 5088  dom cdm 5613   “ cima 5616  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7340  Vtxcvtx 28928  iEdgciedg 28929   ClNeighbVtx cclnbgr 47816   ISubGr cisubgr 47858   ≃_𝑔𝑟 cgric 47874   GraphLocIso cgrlim 47974

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 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-1o 8379  df-map 8746  df-vtx 28930  df-iedg 28931  df-clnbgr 47817  df-isubgr 47859  df-grim 47876  df-gric 47879  df-grlim 47976

This theorem is referenced by:  grlimprop2  47984  uspgrlim  47990  dfgrlic3  48008