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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 29597 | . . 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 29350 | . 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 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Fincfn 8984 Vtxcvtx 29028 USGraphcusgr 29181 FinUSGraphcfusgr 29348 RegUSGraph crusgr 29589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-iota 6516 df-fv 6571 df-fusgr 29349 df-rusgr 29591 |
This theorem is referenced by: numclwwlk1 30390 numclwwlk3 30414 numclwwlk5 30417 numclwwlk7lem 30418 numclwwlk6 30419 frgrreggt1 30422 |
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