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| Mirrors > Home > MPE Home > Th. List > Mathboxes > usgrlimprop | Structured version Visualization version GIF version | ||
| Description: Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.) |
| 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} |
| 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))))) |
| 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 47932 | . 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))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3060 {crab 3435 ⊆ wss 3950 “ cima 5686 –1-1-onto→wf1o 6558 ‘cfv 6559 (class class class)co 7429 Vtxcvtx 29003 Edgcedg 29054 USPGraphcuspgr 29155 ClNeighbVtx cclnbgr 47778 GraphLocIso cgrlim 47916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-1st 8010 df-2nd 8011 df-1o 8502 df-map 8864 df-vtx 29005 df-iedg 29006 df-edg 29055 df-uspgr 29157 df-clnbgr 47779 df-isubgr 47820 df-grim 47837 df-gric 47840 df-grlim 47918 |
| This theorem is referenced by: grlimgrtri 47936 |
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