Theorem frgr2wwlk1 27193

Description: In a friendship graph, there is exactly one walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 4-Jan-2022.)

Hypothesis

Ref	Expression
frgr2wwlkeu.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
frgr2wwlk1	|- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( # ` ( A ( 2 WWalksNOn G ) B ) ) = 1 )

Proof of Theorem frgr2wwlk1

Dummy variables c d t w x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgr2wwlkeu.v	. . . 4 |- V = (Vtx ` G )
2	1	frgr2wwlkn0 27192	. . 3 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( A ( 2 WWalksNOn G ) B ) =/= (/) )
3	1	elwwlks2ons3 26848	. . . . . . . 8 |- ( ( G e. FriendGraph /\ A e. V /\ B e. V ) -> ( w e. ( A ( 2 WWalksNOn G ) B ) <-> E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
4	3	3expb 1266	. . . . . . 7 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) ) -> ( w e. ( A ( 2 WWalksNOn G ) B ) <-> E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
5	4	3adant3 1081	. . . . . 6 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( w e. ( A ( 2 WWalksNOn G ) B ) <-> E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
6	1	elwwlks2ons3 26848	. . . . . . . 8 |- ( ( G e. FriendGraph /\ A e. V /\ B e. V ) -> ( t e. ( A ( 2 WWalksNOn G ) B ) <-> E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
7	6	3expb 1266	. . . . . . 7 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) ) -> ( t e. ( A ( 2 WWalksNOn G ) B ) <-> E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
8	7	3adant3 1081	. . . . . 6 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( t e. ( A ( 2 WWalksNOn G ) B ) <-> E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
9	5, 8	anbi12d 747	. . . . 5 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( w e. ( A ( 2 WWalksNOn G ) B ) /\ t e. ( A ( 2 WWalksNOn G ) B ) ) <-> ( E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) /\ E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) ) )
10	1	frgr2wwlkeu 27191	. . . . . 6 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) )
11		s3eq2 13615	. . . . . . . . . . . . . 14 |- ( x = y -> <" A x B "> = <" A y B "> )
12	11	eleq1d 2686	. . . . . . . . . . . . 13 |- ( x = y -> ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) <-> <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
13	12	reu4 3400	. . . . . . . . . . . 12 |- ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) <-> ( E. x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ A. x e. V A. y e. V ( ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> x = y ) ) )
14		s3eq2 13615	. . . . . . . . . . . . . . . . . 18 |- ( x = d -> <" A x B "> = <" A d B "> )
15	14	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( x = d -> ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) <-> <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
16	15	anbi1d 741	. . . . . . . . . . . . . . . 16 |- ( x = d -> ( ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) <-> ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
17		equequ1 1952	. . . . . . . . . . . . . . . 16 |- ( x = d -> ( x = y <-> d = y ) )
18	16, 17	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( x = d -> ( ( ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> x = y ) <-> ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = y ) ) )
19		s3eq2 13615	. . . . . . . . . . . . . . . . . 18 |- ( y = c -> <" A y B "> = <" A c B "> )
20	19	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( y = c -> ( <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) <-> <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
21	20	anbi2d 740	. . . . . . . . . . . . . . . 16 |- ( y = c -> ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) <-> ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) )
22		equequ2 1953	. . . . . . . . . . . . . . . 16 |- ( y = c -> ( d = y <-> d = c ) )
23	21, 22	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( y = c -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = y ) <-> ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) ) )
24	18, 23	rspc2va 3323	. . . . . . . . . . . . . 14 |- ( ( ( d e. V /\ c e. V ) /\ A. x e. V A. y e. V ( ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> x = y ) ) -> ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) )
25		pm3.35 611	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) /\ ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) ) -> d = c )
26		s3eq2 13615	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( c = d -> <" A c B "> = <" A d B "> )
27	26	equcoms 1947	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( d = c -> <" A c B "> = <" A d B "> )
28	27	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( d = c /\ ( t = <" A c B "> /\ w = <" A d B "> ) ) -> <" A c B "> = <" A d B "> )
29		eqeq12 2635	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( t = <" A c B "> /\ w = <" A d B "> ) -> ( t = w <-> <" A c B "> = <" A d B "> ) )
30	29	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( d = c /\ ( t = <" A c B "> /\ w = <" A d B "> ) ) -> ( t = w <-> <" A c B "> = <" A d B "> ) )
31	28, 30	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( d = c /\ ( t = <" A c B "> /\ w = <" A d B "> ) ) -> t = w )
32	31	equcomd 1946	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( d = c /\ ( t = <" A c B "> /\ w = <" A d B "> ) ) -> w = t )
33	32	ex 450	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( d = c -> ( ( t = <" A c B "> /\ w = <" A d B "> ) -> w = t ) )
34	25, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) /\ ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) ) -> ( ( t = <" A c B "> /\ w = <" A d B "> ) -> w = t ) )
35	34	ex 450	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> ( ( t = <" A c B "> /\ w = <" A d B "> ) -> w = t ) ) )
36	35	com23 86	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( t = <" A c B "> /\ w = <" A d B "> ) -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> w = t ) ) )
37	36	exp4b 632	. . . . . . . . . . . . . . . . . . 19 |- ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( t = <" A c B "> -> ( w = <" A d B "> -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> w = t ) ) ) ) )
38	37	com13 88	. . . . . . . . . . . . . . . . . 18 |- ( t = <" A c B "> -> ( <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( w = <" A d B "> -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> w = t ) ) ) ) )
39	38	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( w = <" A d B "> -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> w = t ) ) ) )
40	39	com13 88	. . . . . . . . . . . . . . . 16 |- ( w = <" A d B "> -> ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> w = t ) ) ) )
41	40	imp 445	. . . . . . . . . . . . . . 15 |- ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> w = t ) ) )
42	41	com13 88	. . . . . . . . . . . . . 14 |- ( ( ( <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> d = c ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) )
43	24, 42	syl 17	. . . . . . . . . . . . 13 |- ( ( ( d e. V /\ c e. V ) /\ A. x e. V A. y e. V ( ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> x = y ) ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) )
44	43	expcom 451	. . . . . . . . . . . 12 |- ( A. x e. V A. y e. V ( ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" A y B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> x = y ) -> ( ( d e. V /\ c e. V ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) ) )
45	13, 44	simplbiim 659	. . . . . . . . . . 11 |- ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( ( d e. V /\ c e. V ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) ) )
46	45	impl 650	. . . . . . . . . 10 |- ( ( ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ d e. V ) /\ c e. V ) -> ( ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) )
47	46	rexlimdva 3031	. . . . . . . . 9 |- ( ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ d e. V ) -> ( E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) )
48	47	com23 86	. . . . . . . 8 |- ( ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) /\ d e. V ) -> ( ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) )
49	48	rexlimdva 3031	. . . . . . 7 |- ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) )
50	49	impd 447	. . . . . 6 |- ( E! x e. V <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( ( E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) /\ E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) -> w = t ) )
51	10, 50	syl 17	. . . . 5 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( E. d e. V ( w = <" A d B "> /\ <" A d B "> e. ( A ( 2 WWalksNOn G ) B ) ) /\ E. c e. V ( t = <" A c B "> /\ <" A c B "> e. ( A ( 2 WWalksNOn G ) B ) ) ) -> w = t ) )
52	9, 51	sylbid 230	. . . 4 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( w e. ( A ( 2 WWalksNOn G ) B ) /\ t e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) )
53	52	alrimivv 1856	. . 3 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> A. w A. t ( ( w e. ( A ( 2 WWalksNOn G ) B ) /\ t e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) )
54		eqeuel 3941	. . 3 |- ( ( ( A ( 2 WWalksNOn G ) B ) =/= (/) /\ A. w A. t ( ( w e. ( A ( 2 WWalksNOn G ) B ) /\ t e. ( A ( 2 WWalksNOn G ) B ) ) -> w = t ) ) -> E! w w e. ( A ( 2 WWalksNOn G ) B ) )
55	2, 53, 54	syl2anc 693	. 2 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> E! w w e. ( A ( 2 WWalksNOn G ) B ) )
56		ovex 6678	. . 3 |- ( A ( 2 WWalksNOn G ) B ) e. _V
57		euhash1 13208	. . 3 |- ( ( A ( 2 WWalksNOn G ) B ) e. _V -> ( ( # ` ( A ( 2 WWalksNOn G ) B ) ) = 1 <-> E! w w e. ( A ( 2 WWalksNOn G ) B ) ) )
58	56, 57	mp1i 13	. 2 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( ( # ` ( A ( 2 WWalksNOn G ) B ) ) = 1 <-> E! w w e. ( A ( 2 WWalksNOn G ) B ) ) )
59	55, 58	mpbird 247	1 |- ( ( G e. FriendGraph /\ ( A e. V /\ B e. V ) /\ A =/= B ) -> ( # ` ( A ( 2 WWalksNOn G ) B ) ) = 1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E!weu 2470   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   1c1 9937   2c2 11070   #chash 13117   <"cs3 13587  Vtxcvtx 25874   WWalksNOn cwwlksnon 26719   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-frgr 27121

This theorem is referenced by:  frgr2wsp1  27194