Theorem finrusgrfusgr 29952

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 29951	. . 3 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2	1	anim1i 627	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3		finrusgrfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	isfusgr 29705	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
5	2, 4	sylibr 237	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   class class class wbr 5114  ‘cfv 6543  Fincfn 8952  Vtxcvtx 29383  USGraphcusgr 29536  FinUSGraphcfusgr 29703   RegUSGraph crusgr 29943

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-iota 6499  df-fv 6551  df-fusgr 29704  df-rusgr 29945

This theorem is used by:  numclwwlk1  30749  numclwwlk3  30773  numclwwlk5  30776  numclwwlk7lem  30777  numclwwlk6  30778  frgrreggt1  30781