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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfgrlic3 | Structured version Visualization version GIF version |
Description: Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.) |
Ref | Expression |
---|---|
dfgrlic2.v | ⊢ 𝑉 = (Vtx‘𝐺) |
dfgrlic2.w | ⊢ 𝑊 = (Vtx‘𝐻) |
dfgrlic3.i | ⊢ 𝐼 = (iEdg‘𝐺) |
dfgrlic3.j | ⊢ 𝐽 = (iEdg‘𝐻) |
dfgrlic3.n | ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣) |
dfgrlic3.m | ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣)) |
dfgrlic3.k | ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} |
dfgrlic3.l | ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} |
Ref | Expression |
---|---|
dfgrlic3 | ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgrlic 47899 | . . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅) | |
2 | n0 4358 | . . 3 ⊢ ((𝐺 GraphLocIso 𝐻) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻)) | |
3 | 1, 2 | bitri 275 | . 2 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻)) |
4 | dfgrlic2.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | dfgrlic2.w | . . . . 5 ⊢ 𝑊 = (Vtx‘𝐻) | |
6 | dfgrlic3.n | . . . . 5 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣) | |
7 | dfgrlic3.m | . . . . 5 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣)) | |
8 | dfgrlic3.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
9 | dfgrlic3.j | . . . . 5 ⊢ 𝐽 = (iEdg‘𝐻) | |
10 | dfgrlic3.k | . . . . 5 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} | |
11 | dfgrlic3.l | . . . . 5 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} | |
12 | 4, 5, 6, 7, 8, 9, 10, 11 | isgrlim2 47885 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝑓 ∈ V) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
13 | 12 | el3v3 3486 | . . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
14 | 13 | exbidv 1918 | . 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
15 | 3, 14 | bitrid 283 | 1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 {crab 3432 Vcvv 3477 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 dom cdm 5688 “ cima 5691 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 Vtxcvtx 29027 iEdgciedg 29028 ClNeighbVtx cclnbgr 47742 GraphLocIso cgrlim 47878 ≃𝑙𝑔𝑟 cgrlic 47879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-1o 8504 df-map 8866 df-vtx 29029 df-iedg 29030 df-clnbgr 47743 df-isubgr 47784 df-grim 47801 df-gric 47804 df-grlim 47880 df-grlic 47883 |
This theorem is referenced by: grilcbri2 47906 |
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