Theorem isrgr 26455

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 = (Vtx ` G )
isrgr.d	|- D = (VtxDeg ` G )

Assertion

Ref	Expression
isrgr	|- ( ( G e. W /\ K e. Z ) -> ( 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)   W( v)   Z( v)

Proof of Theorem isrgr

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

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( k = K -> ( k e. NN0* <-> K e. NN0* ) )
2	1	adantl 482	. . . 4 |- ( ( g = G /\ k = K ) -> ( k e. NN0* <-> K e. NN0* ) )
3		fveq2 6191	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4	3	adantr 481	. . . . 5 |- ( ( g = G /\ k = K ) -> (Vtx ` g ) = (Vtx ` G ) )
5		fveq2 6191	. . . . . . . 8 |- ( g = G -> (VtxDeg ` g ) = (VtxDeg ` G ) )
6	5	fveq1d 6193	. . . . . . 7 |- ( g = G -> ( (VtxDeg ` g ) ` v ) = ( (VtxDeg ` G ) ` v ) )
7	6	adantr 481	. . . . . 6 |- ( ( g = G /\ k = K ) -> ( (VtxDeg ` g ) ` v ) = ( (VtxDeg ` G ) ` v ) )
8		simpr 477	. . . . . 6 |- ( ( g = G /\ k = K ) -> k = K )
9	7, 8	eqeq12d 2637	. . . . 5 |- ( ( g = G /\ k = K ) -> ( ( (VtxDeg ` g ) ` v ) = k <-> ( (VtxDeg ` G ) ` v ) = K ) )
10	4, 9	raleqbidv 3152	. . . 4 |- ( ( g = G /\ k = K ) -> ( A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k <-> A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K ) )
11	2, 10	anbi12d 747	. . 3 |- ( ( g = G /\ k = K ) -> ( ( k e. NN0* /\ A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k ) <-> ( K e. NN0* /\ A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K ) ) )
12		df-rgr 26453	. . 3 |- RegGraph = { <. g , k >. | ( k e. NN0* /\ A. v e. (Vtx ` g ) ( (VtxDeg ` g ) ` v ) = k ) }
13	11, 12	brabga 4989	. 2 |- ( ( G e. W /\ K e. Z ) -> ( G RegGraph K <-> ( K e. NN0* /\ A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K ) ) )
14		isrgr.v	. . . . . 6 |- V = (Vtx ` G )
15		isrgr.d	. . . . . . . 8 |- D = (VtxDeg ` G )
16	15	fveq1i 6192	. . . . . . 7 |- ( D ` v ) = ( (VtxDeg ` G ) ` v )
17	16	eqeq1i 2627	. . . . . 6 |- ( ( D ` v ) = K <-> ( (VtxDeg ` G ) ` v ) = K )
18	14, 17	raleqbii 2990	. . . . 5 |- ( A. v e. V ( D ` v ) = K <-> A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K )
19	18	bicomi 214	. . . 4 |- ( A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K <-> A. v e. V ( D ` v ) = K )
20	19	a1i 11	. . 3 |- ( ( G e. W /\ K e. Z ) -> ( A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K <-> A. v e. V ( D ` v ) = K ) )
21	20	anbi2d 740	. 2 |- ( ( G e. W /\ K e. Z ) -> ( ( K e. NN0* /\ A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = K ) <-> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) ) )
22	13, 21	bitrd 268	1 |- ( ( G e. W /\ K e. Z ) -> ( G RegGraph K <-> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` 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-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:  rgrprop  26456  isrusgr0  26462  0edg0rgr  26468  0vtxrgr  26472  rgrprcx  26488