Theorem isgrlim2 48471

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 48470	. 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 2746	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) = 𝑁
6	5	oveq2i 7371	. . . . . . 7 ⊢ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr 𝑁)
7		isgrlim2.m	. . . . . . . . 9 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
8	7	eqcomi 2746	. . . . . . . 8 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝑣)) = 𝑀
9	8	oveq2i 7371	. . . . . . 7 ⊢ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) = (𝐻 ISubGr 𝑀)
10	6, 9	breq12i 5095	. . . . . 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 48420	. . . . . 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 1133	. . . . 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 3161	. . 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 631	. 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  {crab 3390   ⊆ wss 3890   class class class wbr 5086  dom cdm 5624   “ cima 5627  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  Vtxcvtx 29079  iEdgciedg 29080   ClNeighbVtx cclnbgr 48306   ISubGr cisubgr 48348   ≃_𝑔𝑟 cgric 48364   GraphLocIso cgrlim 48464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-1o 8398  df-map 8768  df-vtx 29081  df-iedg 29082  df-clnbgr 48307  df-isubgr 48349  df-grim 48366  df-gric 48369  df-grlim 48466

This theorem is referenced by:  grlimprop2  48474  uspgrlim  48480  dfgrlic3  48498