Theorem s2elclwwlknon2 16431

Description: Sufficient conditions of a doubleton word to represent a closed walk on vertex X of length 2. (Contributed by AV, 11-May-2022.)

Hypotheses

Ref	Expression
clwwlknon2.c	|- C = (ClWWalksNOn ` G )
clwwlknon2x.v	|- V = (Vtx ` G )
clwwlknon2x.e	|- E = (Edg ` G )

Assertion

Ref	Expression
s2elclwwlknon2	|- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. ( X C 2 ) )

Proof of Theorem s2elclwwlknon2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		s2cl 11477	. . 3 |- ( ( X e. V /\ Y e. V ) -> <" X Y "> e. Word V )
2	1	3adant3 1044	. 2 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. Word V )
3		s2leng 11481	. . . 4 |- ( ( X e. V /\ Y e. V ) -> (♯ ` <" X Y "> ) = 2 )
4	3	3adant3 1044	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> (♯ ` <" X Y "> ) = 2 )
5		s2fv0g 11479	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( <" X Y "> ` 0 ) = X )
6		s2fv1g 11480	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( <" X Y "> ` 1 ) = Y )
7	5, 6	preq12d 3776	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } = { X , Y } )
8	7	eqcomd 2238	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> { X , Y } = { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } )
9	8	eleq1d 2301	. . . 4 |- ( ( X e. V /\ Y e. V ) -> ( { X , Y } e. E <-> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E ) )
10	9	biimp3a 1382	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E )
11	5	3adant3 1044	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> ( <" X Y "> ` 0 ) = X )
12	4, 10, 11	3jca 1204	. 2 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> ( (♯ ` <" X Y "> ) = 2 /\ { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E /\ ( <" X Y "> ` 0 ) = X ) )
13		fveqeq2 5679	. . . 4 |- ( w = <" X Y "> -> ( (♯ ` w ) = 2 <-> (♯ ` <" X Y "> ) = 2 ) )
14		fveq1 5669	. . . . . 6 |- ( w = <" X Y "> -> ( w ` 0 ) = ( <" X Y "> ` 0 ) )
15		fveq1 5669	. . . . . 6 |- ( w = <" X Y "> -> ( w ` 1 ) = ( <" X Y "> ` 1 ) )
16	14, 15	preq12d 3776	. . . . 5 |- ( w = <" X Y "> -> { ( w ` 0 ) , ( w ` 1 ) } = { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } )
17	16	eleq1d 2301	. . . 4 |- ( w = <" X Y "> -> ( { ( w ` 0 ) , ( w ` 1 ) } e. E <-> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E ) )
18	14	eqeq1d 2241	. . . 4 |- ( w = <" X Y "> -> ( ( w ` 0 ) = X <-> ( <" X Y "> ` 0 ) = X ) )
19	13, 17, 18	3anbi123d 1349	. . 3 |- ( w = <" X Y "> -> ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) <-> ( (♯ ` <" X Y "> ) = 2 /\ { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E /\ ( <" X Y "> ` 0 ) = X ) ) )
20		clwwlknon2.c	. . . 4 |- C = (ClWWalksNOn ` G )
21		clwwlknon2x.v	. . . 4 |- V = (Vtx ` G )
22		clwwlknon2x.e	. . . 4 |- E = (Edg ` G )
23	20, 21, 22	clwwlknon2x 16430	. . 3 |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }
24	19, 23	elrab2 2976	. 2 |- ( <" X Y "> e. ( X C 2 ) <-> ( <" X Y "> e. Word V /\ ( (♯ ` <" X Y "> ) = 2 /\ { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E /\ ( <" X Y "> ` 0 ) = X ) ) )
25	2, 12, 24	sylanbrc 417	1 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. ( X C 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   {cpr 3690   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   2c2 9288  ♯chash 11138  Word cword 11224   <"cs2 11441  Vtxcvtx 16007  Edgcedg 16052  ClWWalksNOncclwwlknon 16421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-map 6884  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-lsw 11270  df-concat 11279  df-s1 11304  df-s2 11448  df-ndx 13215  df-slot 13216  df-base 13218  df-vtx 16009  df-clwwlk 16387  df-clwwlkn 16399  df-clwwlknon 16422

This theorem is referenced by: (None)