Theorem finrusgrfusgr 27347

Description: A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)

Hypothesis

Ref	Expression
finrusgrfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
finrusgrfusgr	⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)

Proof of Theorem finrusgrfusgr

Step	Hyp	Ref	Expression
1		rusgrusgr 27346	. . 3 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2	1	anim1i 616	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3		finrusgrfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	isfusgr 27100	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
5	2, 4	sylibr 236	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   class class class wbr 5066  ‘cfv 6355  Fincfn 8509  Vtxcvtx 26781  USGraphcusgr 26934  FinUSGraphcfusgr 27098   RegUSGraph crusgr 27338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-iota 6314  df-fv 6363  df-fusgr 27099  df-rusgr 27340

This theorem is referenced by:  numclwwlk1  28140  numclwwlk3  28164  numclwwlk5  28167  numclwwlk7lem  28168  numclwwlk6  28169  frgrreggt1  28172