Theorem finrusgrfusgr 27835

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 27834	. . 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 27588	. 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 1539   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  Fincfn 8691  Vtxcvtx 27269  USGraphcusgr 27422  FinUSGraphcfusgr 27586   RegUSGraph crusgr 27826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-iota 6376  df-fv 6426  df-fusgr 27587  df-rusgr 27828

This theorem is referenced by:  numclwwlk1  28626  numclwwlk3  28650  numclwwlk5  28653  numclwwlk7lem  28654  numclwwlk6  28655  frgrreggt1  28658