Theorem usgrlimprop 48469

Description: Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.)

Hypotheses

Ref	Expression
usgrlimprop.v	⊢ 𝑉 = (Vtx‘𝐺)
usgrlimprop.w	⊢ 𝑊 = (Vtx‘𝐻)
usgrlimprop.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
usgrlimprop.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
usgrlimprop.i	⊢ 𝐼 = (Edg‘𝐺)
usgrlimprop.j	⊢ 𝐽 = (Edg‘𝐻)
usgrlimprop.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
usgrlimprop.l	⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}

Assertion

Ref	Expression
usgrlimprop	⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 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 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))

Distinct variable groups:   𝑓,𝐹,𝑣   𝑒,𝐺,𝑓,𝑔,𝑣,𝑥   𝑒,𝐻,𝑓,𝑔,𝑣,𝑥   𝑥,𝐼   𝑥,𝐽   𝑒,𝐾,𝑔,𝑥   𝑔,𝐿,𝑥   𝑒,𝑀,𝑓,𝑔,𝑥   𝑒,𝑁,𝑓,𝑔,𝑥   𝑣,𝑉

Allowed substitution hints:   𝐹(𝑥,𝑒,𝑔)   𝐼(𝑣,𝑒,𝑓,𝑔)   𝐽(𝑣,𝑒,𝑓,𝑔)   𝐾(𝑣,𝑓)   𝐿(𝑣,𝑒,𝑓)   𝑀(𝑣)   𝑁(𝑣)   𝑉(𝑥,𝑒,𝑓,𝑔)   𝑊(𝑥,𝑣,𝑒,𝑓,𝑔)

Proof of Theorem usgrlimprop

Step	Hyp	Ref	Expression
1		simp3 1139	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
2		usgrlimprop.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
3		usgrlimprop.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐻)
4		usgrlimprop.n	. . 3 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
5		usgrlimprop.m	. . 3 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
6		usgrlimprop.i	. . 3 ⊢ 𝐼 = (Edg‘𝐺)
7		usgrlimprop.j	. . 3 ⊢ 𝐽 = (Edg‘𝐻)
8		usgrlimprop.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
9		usgrlimprop.l	. . 3 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
10	2, 3, 4, 5, 6, 7, 8, 9	uspgrlim 48468	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (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 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))))))
11	1, 10	mpbid 232	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 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 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  {crab 3389   ⊆ wss 3889   “ cima 5634  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  Vtxcvtx 29065  Edgcedg 29116  USPGraphcuspgr 29217   ClNeighbVtx cclnbgr 48294   GraphLocIso cgrlim 48452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-1o 8405  df-map 8775  df-vtx 29067  df-iedg 29068  df-edg 29117  df-uspgr 29219  df-clnbgr 48295  df-isubgr 48337  df-grim 48354  df-gric 48357  df-grlim 48454

This theorem is referenced by:  grlimedgclnbgr  48471  grlimgrtri  48479