Theorem isfrgr 27122

Description: The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
isfrgr.v	|- V = (Vtx ` G )
isfrgr.e	|- E = (Edg ` G )

Assertion

Ref	Expression
isfrgr	|- ( G e. U -> ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) )

Distinct variable groups:   k, l, x, G   k, V, l, x

Allowed substitution hints:   U( x, k, l)   E( x, k, l)

Proof of Theorem isfrgr

Dummy variables e h g v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frgr 27121	. . 3 |- FriendGraph = { g | ( g e. USGraph /\ [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e ) }
2	1	eleq2i 2693	. 2 |- ( G e. FriendGraph <-> G e. { g | ( g e. USGraph /\ [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e ) } )
3		eleq1 2689	. . . 4 |- ( h = G -> ( h e. USGraph <-> G e. USGraph ) )
4		fveq2 6191	. . . . . 6 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
5		isfrgr.v	. . . . . 6 |- V = (Vtx ` G )
6	4, 5	syl6eqr 2674	. . . . 5 |- ( h = G -> (Vtx ` h ) = V )
7	6	difeq1d 3727	. . . . . 6 |- ( h = G -> ( (Vtx ` h ) \ { k } ) = ( V \ { k } ) )
8		reueq1 3140	. . . . . . . 8 |- ( (Vtx ` h ) = V -> ( E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) <-> E! x e. V { { x , k } , { x , l } } C_ (Edg ` h ) ) )
9	6, 8	syl 17	. . . . . . 7 |- ( h = G -> ( E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) <-> E! x e. V { { x , k } , { x , l } } C_ (Edg ` h ) ) )
10		fveq2 6191	. . . . . . . . . 10 |- ( h = G -> (Edg ` h ) = (Edg ` G ) )
11		isfrgr.e	. . . . . . . . . 10 |- E = (Edg ` G )
12	10, 11	syl6eqr 2674	. . . . . . . . 9 |- ( h = G -> (Edg ` h ) = E )
13	12	sseq2d 3633	. . . . . . . 8 |- ( h = G -> ( { { x , k } , { x , l } } C_ (Edg ` h ) <-> { { x , k } , { x , l } } C_ E ) )
14	13	reubidv 3126	. . . . . . 7 |- ( h = G -> ( E! x e. V { { x , k } , { x , l } } C_ (Edg ` h ) <-> E! x e. V { { x , k } , { x , l } } C_ E ) )
15	9, 14	bitrd 268	. . . . . 6 |- ( h = G -> ( E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) <-> E! x e. V { { x , k } , { x , l } } C_ E ) )
16	7, 15	raleqbidv 3152	. . . . 5 |- ( h = G -> ( A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) <-> A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) )
17	6, 16	raleqbidv 3152	. . . 4 |- ( h = G -> ( A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) <-> A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) )
18	3, 17	anbi12d 747	. . 3 |- ( h = G -> ( ( h e. USGraph /\ A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) )
19		eleq1 2689	. . . . 5 |- ( g = h -> ( g e. USGraph <-> h e. USGraph ) )
20		fvexd 6203	. . . . . 6 |- ( g = h -> (Vtx ` g ) e. _V )
21		fveq2 6191	. . . . . 6 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
22		fvexd 6203	. . . . . . 7 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (Edg ` g ) e. _V )
23		fveq2 6191	. . . . . . . 8 |- ( g = h -> (Edg ` g ) = (Edg ` h ) )
24	23	adantr 481	. . . . . . 7 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (Edg ` g ) = (Edg ` h ) )
25		simpr 477	. . . . . . . . 9 |- ( ( g = h /\ v = (Vtx ` h ) ) -> v = (Vtx ` h ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> v = (Vtx ` h ) )
27		difeq1 3721	. . . . . . . . . 10 |- ( v = (Vtx ` h ) -> ( v \ { k } ) = ( (Vtx ` h ) \ { k } ) )
28	27	ad2antlr 763	. . . . . . . . 9 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( v \ { k } ) = ( (Vtx ` h ) \ { k } ) )
29		reueq1 3140	. . . . . . . . . . 11 |- ( v = (Vtx ` h ) -> ( E! x e. v { { x , k } , { x , l } } C_ e <-> E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ e ) )
30	29	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( E! x e. v { { x , k } , { x , l } } C_ e <-> E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ e ) )
31		simpr 477	. . . . . . . . . . . 12 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> e = (Edg ` h ) )
32	31	sseq2d 3633	. . . . . . . . . . 11 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( { { x , k } , { x , l } } C_ e <-> { { x , k } , { x , l } } C_ (Edg ` h ) ) )
33	32	reubidv 3126	. . . . . . . . . 10 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ e <-> E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) )
34	30, 33	bitrd 268	. . . . . . . . 9 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( E! x e. v { { x , k } , { x , l } } C_ e <-> E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) )
35	28, 34	raleqbidv 3152	. . . . . . . 8 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e <-> A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) )
36	26, 35	raleqbidv 3152	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (Edg ` h ) ) -> ( A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e <-> A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) )
37	22, 24, 36	sbcied2 3473	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e <-> A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) )
38	20, 21, 37	sbcied2 3473	. . . . 5 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e <-> A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) )
39	19, 38	anbi12d 747	. . . 4 |- ( g = h -> ( ( g e. USGraph /\ [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e ) <-> ( h e. USGraph /\ A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) ) )
40	39	cbvabv 2747	. . 3 |- { g | ( g e. USGraph /\ [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e ) } = { h | ( h e. USGraph /\ A. k e. (Vtx ` h ) A. l e. ( (Vtx ` h ) \ { k } ) E! x e. (Vtx ` h ) { { x , k } , { x , l } } C_ (Edg ` h ) ) }
41	18, 40	elab2g 3353	. 2 |- ( G e. U -> ( G e. { g | ( g e. USGraph /\ [. (Vtx ` g ) / v ]. [. (Edg ` g ) / e ]. A. k e. v A. l e. ( v \ { k } ) E! x e. v { { x , k } , { x , l } } C_ e ) } <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) )
42	2, 41	syl5bb 272	1 |- ( G e. U -> ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E!wreu 2914   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

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-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  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-iota 5851  df-fv 5896  df-frgr 27121

This theorem is referenced by:  frgrusgrfrcond  27123