Theorem finrusgrfusgr 29291

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 29290	. . 3 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
2	1	anim1i 614	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3		finrusgrfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	isfusgr 29044	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
5	2, 4	sylibr 233	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5138  ‘cfv 6533  Fincfn 8935  Vtxcvtx 28725  USGraphcusgr 28878  FinUSGraphcfusgr 29042   RegUSGraph crusgr 29282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-iota 6485  df-fv 6541  df-fusgr 29043  df-rusgr 29284

This theorem is referenced by:  numclwwlk1  30083  numclwwlk3  30107  numclwwlk5  30110  numclwwlk7lem  30111  numclwwlk6  30112  frgrreggt1  30115