Theorem finrusgrfusgr 26461

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 26460	. . 3 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph )
2	1	anim1i 592	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3		finrusgrfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	isfusgr 26210	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
5	2, 4	sylibr 224	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   class class class wbr 4653  ‘cfv 5888  Fincfn 7955  Vtxcvtx 25874   USGraph cusgr 26044   FinUSGraph cfusgr 26208   RegUSGraph crusgr 26452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-fusgr 26209  df-rusgr 26454

This theorem is referenced by:  numclwwlk1  27231  numclwwlk3OLD  27242  numclwwlk3  27243  numclwwlk5  27246  numclwwlk7lem  27247  numclwwlk6  27248  frgrreggt1  27251