Theorem usgrlimprop 49227

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 49226	. 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)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3896   “ cima 5654  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7424  Vtxcvtx 29583  Edgcedg 29634  USPGraphcuspgr 29738   ClNeighbVtx cclnbgr 49052   GraphLocIso cgrlim 49210

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-1o 8484  df-map 8857  df-vtx 29585  df-iedg 29586  df-edg 29635  df-uspgr 29740  df-clnbgr 49053  df-isubgr 49095  df-grim 49112  df-gric 49115  df-grlim 49212

This theorem is used by:  grlimedgclnbgr  49229  grlimgrtri  49237