Theorem clnbgrisubgrgrim 47929

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 2729	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3	2	clnbgrssvtx 47829	. . 3 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ (Vtx‘𝐺)
4	1, 3	eqsstri 3993	. 2 ⊢ 𝑁 ⊆ (Vtx‘𝐺)
5		clnbgrisubgrgrim.m	. . 3 ⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑌)
6		eqid 2729	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
7	6	clnbgrssvtx 47829	. . 3 ⊢ (𝐻 ClNeighbVtx 𝑌) ⊆ (Vtx‘𝐻)
8	5, 7	eqsstri 3993	. 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 47926	. 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 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  {crab 3405   ⊆ wss 3914   class class class wbr 5107  dom cdm 5638   “ cima 5641  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  iEdgciedg 28924   ClNeighbVtx cclnbgr 47816   ISubGr cisubgr 47857   ≃_𝑔𝑟 cgric 47873

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-1o 8434  df-map 8801  df-vtx 28925  df-iedg 28926  df-clnbgr 47817  df-isubgr 47858  df-grim 47875  df-gric 47878

This theorem is referenced by:  isgrlim2  47979