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| Mirrors > Home > MPE Home > Th. List > finrusgrfusgr | Structured version Visualization version GIF version | ||
| Description: A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.) |
| Ref | Expression |
|---|---|
| finrusgrfusgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| Ref | Expression |
|---|---|
| finrusgrfusgr | ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rusgrusgr 29582 | . . 3 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph) | |
| 2 | 1 | anim1i 615 | . 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
| 3 | finrusgrfusgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 4 | 3 | isfusgr 29335 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
| 5 | 2, 4 | sylibr 234 | 1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Fincfn 8985 Vtxcvtx 29013 USGraphcusgr 29166 FinUSGraphcfusgr 29333 RegUSGraph crusgr 29574 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-iota 6514 df-fv 6569 df-fusgr 29334 df-rusgr 29576 |
| This theorem is referenced by: numclwwlk1 30380 numclwwlk3 30404 numclwwlk5 30407 numclwwlk7lem 30408 numclwwlk6 30409 frgrreggt1 30412 |
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