Theorem usgrlimprop 47807

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 47806	. 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 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  {crab 3443   ⊆ wss 3976   “ cima 5698  –1-1-onto→wf1o 6567  ‘cfv 6568  (class class class)co 7443  Vtxcvtx 29023  Edgcedg 29074  USPGraphcuspgr 29175   ClNeighbVtx cclnbgr 47682   GraphLocIso cgrlim 47790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-suc 6396  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8024  df-2nd 8025  df-1o 8516  df-map 8880  df-vtx 29025  df-iedg 29026  df-edg 29075  df-uspgr 29177  df-clnbgr 47683  df-isubgr 47723  df-grim 47738  df-gric 47741  df-grlim 47792

This theorem is referenced by:  grlimgrtri  47810