MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isrgr Structured version   Visualization version   GIF version
Theorem isrgr 28816
Description: The property of a class being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrgr.v 𝑉 = (Vtxβ€˜πΊ)
isrgr.d 𝐷 = (VtxDegβ€˜πΊ)
Assertion
Ref Expression
isrgr ((𝐺 ∈ π‘Š ∧ 𝐾 ∈ 𝑍) β†’ (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 (π·β€˜π‘£) = 𝐾)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)   π‘Š(𝑣)   𝑍(𝑣)
Proof of Theorem isrgr
Dummy variables 𝑔 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2822 . . . . 5 (π‘˜ = 𝐾 β†’ (π‘˜ ∈ β„•0* ↔ 𝐾 ∈ β„•0*))
21adantl 483 . . . 4 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ (π‘˜ ∈ β„•0* ↔ 𝐾 ∈ β„•0*))
3 fveq2 6892 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
43adantr 482 . . . . 5 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
5 fveq2 6892 . . . . . . . 8 (𝑔 = 𝐺 β†’ (VtxDegβ€˜π‘”) = (VtxDegβ€˜πΊ))
65fveq1d 6894 . . . . . . 7 (𝑔 = 𝐺 β†’ ((VtxDegβ€˜π‘”)β€˜π‘£) = ((VtxDegβ€˜πΊ)β€˜π‘£))
76adantr 482 . . . . . 6 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ ((VtxDegβ€˜π‘”)β€˜π‘£) = ((VtxDegβ€˜πΊ)β€˜π‘£))
8 simpr 486 . . . . . 6 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ π‘˜ = 𝐾)
97, 8eqeq12d 2749 . . . . 5 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ (((VtxDegβ€˜π‘”)β€˜π‘£) = π‘˜ ↔ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾))
104, 9raleqbidv 3343 . . . 4 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ (βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = π‘˜ ↔ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾))
112, 10anbi12d 632 . . 3 ((𝑔 = 𝐺 ∧ π‘˜ = 𝐾) β†’ ((π‘˜ ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = π‘˜) ↔ (𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾)))
12 df-rgr 28814 . . 3 RegGraph = {βŸ¨π‘”, π‘˜βŸ© ∣ (π‘˜ ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜π‘”)((VtxDegβ€˜π‘”)β€˜π‘£) = π‘˜)}
1311, 12brabga 5535 . 2 ((𝐺 ∈ π‘Š ∧ 𝐾 ∈ 𝑍) β†’ (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾)))
14 isrgr.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
15 isrgr.d . . . . . . . 8 𝐷 = (VtxDegβ€˜πΊ)
1615fveq1i 6893 . . . . . . 7 (π·β€˜π‘£) = ((VtxDegβ€˜πΊ)β€˜π‘£)
1716eqeq1i 2738 . . . . . 6 ((π·β€˜π‘£) = 𝐾 ↔ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾)
1814, 17raleqbii 3339 . . . . 5 (βˆ€π‘£ ∈ 𝑉 (π·β€˜π‘£) = 𝐾 ↔ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾)
1918bicomi 223 . . . 4 (βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 ↔ βˆ€π‘£ ∈ 𝑉 (π·β€˜π‘£) = 𝐾)
2019a1i 11 . . 3 ((𝐺 ∈ π‘Š ∧ 𝐾 ∈ 𝑍) β†’ (βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 ↔ βˆ€π‘£ ∈ 𝑉 (π·β€˜π‘£) = 𝐾))
2120anbi2d 630 . 2 ((𝐺 ∈ π‘Š ∧ 𝐾 ∈ 𝑍) β†’ ((𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) ↔ (𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 (π·β€˜π‘£) = 𝐾)))
2213, 21bitrd 279 1 ((𝐺 ∈ π‘Š ∧ 𝐾 ∈ 𝑍) β†’ (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 (π·β€˜π‘£) = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   class class class wbr 5149  β€˜cfv 6544  β„•0*cxnn0 12544  Vtxcvtx 28256  VtxDegcvtxdg 28722   RegGraph crgr 28812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-rgr 28814
This theorem is referenced by:  rgrprop  28817  isrusgr0  28823  0edg0rgr  28829  0vtxrgr  28833  rgrprcx  28849
  Copyright terms: Public domain W3C validator