Theorem usgrlimprop 48024

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 1138	. 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 48023	. 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 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897   “ cima 5614  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341  Vtxcvtx 28969  Edgcedg 29020  USPGraphcuspgr 29121   ClNeighbVtx cclnbgr 47849   GraphLocIso cgrlim 48007

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-1o 8380  df-map 8747  df-vtx 28971  df-iedg 28972  df-edg 29021  df-uspgr 29123  df-clnbgr 47850  df-isubgr 47892  df-grim 47909  df-gric 47912  df-grlim 48009

This theorem is referenced by:  grlimedgclnbgr  48026  grlimgrtri  48034