Theorem usgrlimprop 48758

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 1156	. 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 48757	. 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 235	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 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  {crab 3416   ⊆ wss 3905   “ cima 5664  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346  Edgcedg 29397  USPGraphcuspgr 29498   ClNeighbVtx cclnbgr 48583   GraphLocIso cgrlim 48741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-1o 8449  df-map 8822  df-vtx 29348  df-iedg 29349  df-edg 29398  df-uspgr 29500  df-clnbgr 48584  df-isubgr 48626  df-grim 48643  df-gric 48646  df-grlim 48743

This theorem is referenced by:  grlimedgclnbgr  48760  grlimgrtri  48768