Theorem rgrprop 26456

Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)

Hypotheses

Ref	Expression
isrgr.v	|- V = (Vtx ` G )
isrgr.d	|- D = (VtxDeg ` G )

Assertion

Ref	Expression
rgrprop	|- ( G RegGraph K -> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) )

Distinct variable groups:   v, G   v, K

Allowed substitution hints:   D( v)   V( v)

Proof of Theorem rgrprop

Dummy variables g k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rgr 26453	. . . 4 |- RegGraph = { <. g , k >. | ( k e. NN0* /\ A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k ) }
2	1	breqi 4659	. . 3 |- ( G RegGraph K <-> G { <. g , k >. | ( k e. NN0* /\ A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k ) } K )
3		brabv 6699	. . 3 |- ( G { <. g , k >. | ( k e. NN0* /\ A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k ) } K -> ( G e. _V /\ K e. _V ) )
4	2, 3	sylbi 207	. 2 |- ( G RegGraph K -> ( G e. _V /\ K e. _V ) )
5		isrgr.v	. . . 4 |- V = (Vtx ` G )
6		isrgr.d	. . . 4 |- D = (VtxDeg ` G )
7	5, 6	isrgr 26455	. . 3 |- ( ( G e. _V /\ K e. _V ) -> ( G RegGraph K <-> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) ) )
8	7	biimpd 219	. 2 |- ( ( G e. _V /\ K e. _V ) -> ( G RegGraph K -> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) ) )
9	4, 8	mpcom 38	1 |- ( G RegGraph K -> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   {copab 4712   ` cfv 5888  NN0*cxnn0 11363  Vtxcvtx 25874  VtxDegcvtxdg 26361   RegGraph crgr 26451

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-ral 2917  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-rgr 26453

This theorem is referenced by:  rusgrprop0  26463  uhgr0edg0rgrb  26470  frrusgrord  27205