Theorem clnbgrisubgrgrim 48188

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 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3	2	clnbgrssvtx 48087	. . 3 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ (Vtx‘𝐺)
4	1, 3	eqsstri 3980	. 2 ⊢ 𝑁 ⊆ (Vtx‘𝐺)
5		clnbgrisubgrgrim.m	. . 3 ⊢ 𝑀 = (𝐻 ClNeighbVtx 𝑌)
6		eqid 2736	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
7	6	clnbgrssvtx 48087	. . 3 ⊢ (𝐻 ClNeighbVtx 𝑌) ⊆ (Vtx‘𝐻)
8	5, 7	eqsstri 3980	. 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 48185	. 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 3051  {crab 3399   ⊆ wss 3901   class class class wbr 5098  dom cdm 5624   “ cima 5627  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  iEdgciedg 29070   ClNeighbVtx cclnbgr 48074   ISubGr cisubgr 48116   ≃_𝑔𝑟 cgric 48132

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  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 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-1o 8397  df-map 8765  df-vtx 29071  df-iedg 29072  df-clnbgr 48075  df-isubgr 48117  df-grim 48134  df-gric 48137

This theorem is referenced by:  isgrlim2  48239