Theorem isgrlim2 48789

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 48788	. 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 2775	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) = 𝑁
6	5	oveq2i 7434	. . . . . . 7 ⊢ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr 𝑁)
7		isgrlim2.m	. . . . . . . . 9 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
8	7	eqcomi 2775	. . . . . . . 8 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝑣)) = 𝑀
9	8	oveq2i 7434	. . . . . . 7 ⊢ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) = (𝐻 ISubGr 𝑀)
10	6, 9	breq12i 5123	. . . . . 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 48738	. . . . . 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 1150	. . . . 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 282	. . . 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 3191	. . 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 642	. 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 282	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)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  {crab 3419   ⊆ wss 3908   class class class wbr 5114  dom cdm 5666   “ cima 5669  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Vtxcvtx 29383  iEdgciedg 29384   ClNeighbVtx cclnbgr 48624   ISubGr cisubgr 48666   ≃_𝑔𝑟 cgric 48682   GraphLocIso cgrlim 48782

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-1o 8462  df-map 8835  df-vtx 29385  df-iedg 29386  df-clnbgr 48625  df-isubgr 48667  df-grim 48684  df-gric 48687  df-grlim 48784

This theorem is used by:  grlimprop2  48792  uspgrlim  48798  dfgrlic3  48816