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Mathbox for Alexander van der Vekens |
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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 47530 | . . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅) | |
2 | n0 4346 | . . 3 ⊢ ((𝐺 GraphLocIso 𝐻) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻)) | |
3 | 1, 2 | bitri 274 | . 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 47525 | . . . 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 3471 | . . 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 1917 | . 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 282 | 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 205 ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 {crab 3419 Vcvv 3462 ⊆ wss 3946 ∅c0 4322 class class class wbr 5145 dom cdm 5674 “ cima 5677 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7416 Vtxcvtx 28929 iEdgciedg 28930 ClNeighbVtx cclnbgr 47426 GraphLocIso cgrlim 47518 ≃𝑙𝑔𝑟 cgrlic 47519 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-1o 8488 df-map 8849 df-vtx 28931 df-iedg 28932 df-clnbgr 47427 df-isubgr 47464 df-grim 47479 df-gric 47482 df-grlim 47520 df-grlic 47523 |
This theorem is referenced by: grilcbri2 47537 |
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