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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 1136 | . 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 47817 | . 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 1085 = wceq 1535 ∃wex 1774 ∈ wcel 2104 ∀wral 3057 {crab 3432 ⊆ wss 3963 “ cima 5686 –1-1-onto→wf1o 6557 ‘cfv 6558 (class class class)co 7425 Vtxcvtx 29009 Edgcedg 29060 USPGraphcuspgr 29161 ClNeighbVtx cclnbgr 47693 GraphLocIso cgrlim 47801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 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 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-1o 8499 df-map 8861 df-vtx 29011 df-iedg 29012 df-edg 29061 df-uspgr 29163 df-clnbgr 47694 df-isubgr 47734 df-grim 47749 df-gric 47752 df-grlim 47803 |
This theorem is referenced by: grlimgrtri 47821 |
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