| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| 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 47964 | . . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅) | |
| 2 | n0 4353 | . . 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 47950 | . . . 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 3489 | . . 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 1921 | . 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 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 dom cdm 5685 “ cima 5688 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 iEdgciedg 29014 ClNeighbVtx cclnbgr 47805 GraphLocIso cgrlim 47943 ≃𝑙𝑔𝑟 cgrlic 47944 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-map 8868 df-vtx 29015 df-iedg 29016 df-clnbgr 47806 df-isubgr 47847 df-grim 47864 df-gric 47867 df-grlim 47945 df-grlic 47948 |
| This theorem is referenced by: grilcbri2 47971 |
| Copyright terms: Public domain | W3C validator |