Theorem clnbgrisubgrgrim 48094

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 2733	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3	2	clnbgrssvtx 47993	. . 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 2733	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
7	6	clnbgrssvtx 47993	. . 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 48091	. 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 705	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))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  {crab 3396   ⊆ wss 3898   class class class wbr 5095  dom cdm 5621   “ cima 5624  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7355  Vtxcvtx 28995  iEdgciedg 28996   ClNeighbVtx cclnbgr 47980   ISubGr cisubgr 48022   ≃_𝑔𝑟 cgric 48038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-1o 8394  df-map 8761  df-vtx 28997  df-iedg 28998  df-clnbgr 47981  df-isubgr 48023  df-grim 48040  df-gric 48043

This theorem is referenced by:  isgrlim2  48145