Theorem clnbgrisubgrgrim 48974

Description: Isomorphic subgraphs induced by closed neighborhoods of vertices of two graphs. (Contributed by AV, 29-May-2025.)

Hypotheses

Ref	Expression
clnbgrisubgrgrim.i	⊢ 𝐼 = (iEdg‘𝐺)
clnbgrisubgrgrim.j	⊢ 𝐽 = (iEdg‘𝐻)
clnbgrisubgrgrim.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)
clnbgrisubgrgrim.m	⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑌)
clnbgrisubgrgrim.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
clnbgrisubgrgrim.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

Assertion

Ref	Expression
clnbgrisubgrgrim	⊢ ((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) → ((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))))))

Distinct variable groups:   𝑓,𝐺,𝑔,𝑖   𝑥,𝐺   𝑓,𝐻,𝑔,𝑖   𝑥,𝐻   𝑥,𝐼   𝑥,𝐽   𝑓,𝑀,𝑔,𝑖   𝑥,𝑀   𝑓,𝑁,𝑔,𝑖   𝑥,𝑁   𝑇,𝑓,𝑔,𝑖   𝑈,𝑓,𝑔,𝑖   𝑖,𝐾   𝑖,𝐿

Allowed substitution hints:   𝑇(𝑥)   𝑈(𝑥)   𝐼(𝑓, 𝑔, 𝑖)   𝐽(𝑓, 𝑔, 𝑖)   𝐾(𝑥, 𝑓, 𝑔)   𝐿(𝑥, 𝑓, 𝑔)   𝑋(𝑥, 𝑓, 𝑔, 𝑖)   𝑌(𝑥, 𝑓, 𝑔, 𝑖)

Proof of Theorem clnbgrisubgrgrim

Step	Hyp	Ref	Expression
1		clnbgrisubgrgrim.n	. . 3 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)
2		eqid 2761	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3	2	clnbgrssvtx 48873	. . 3 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ (Vtx‘𝐺)
4	1, 3	eqsstri 3977	. 2 ⊢ 𝑁 ⊆ (Vtx‘𝐺)
5		clnbgrisubgrgrim.m	. . 3 ⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑌)
6		eqid 2761	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
7	6	clnbgrssvtx 48873	. . 3 ⊢ (𝐻 ClNeighbVtx 𝑌) ⊆ (Vtx‘𝐻)
8	5, 7	eqsstri 3977	. 2 ⊢ 𝑀 ⊆ (Vtx‘𝐻)
9		clnbgrisubgrgrim.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
10		clnbgrisubgrgrim.j	. . 3 ⊢ 𝐽 = (iEdg‘𝐻)
11		clnbgrisubgrgrim.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
12		clnbgrisubgrgrim.l	. . 3 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
13	2, 6, 9, 10, 11, 12	isubgrgrim 48971	. 2 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ (Vtx‘𝐺) ∧ 𝑀 ⊆ (Vtx‘𝐻))) → ((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))))))
14	4, 8, 13	mpanr12 718	1 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) → ((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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899   class class class wbr 5103  dom cdm 5651   “ cima 5654  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412  Vtxcvtx 29556  iEdgciedg 29557   ClNeighbVtx cclnbgr 48860   ISubGr cisubgr 48902   ≃_𝑔𝑟 cgric 48918

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 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-1o 8460  df-map 8833  df-vtx 29558  df-iedg 29559  df-clnbgr 48861  df-isubgr 48903  df-grim 48920  df-gric 48923

This theorem is used by:  isgrlim2  49025