Theorem s2elclwwlknon2 16286

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 11365	. . 3 |- ( ( X e. V /\ Y e. V ) -> <" X Y "> e. Word V )
2	1	3adant3 1043	. 2 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. Word V )
3		s2leng 11369	. . . 4 |- ( ( X e. V /\ Y e. V ) -> (♯ ` <" X Y "> ) = 2 )
4	3	3adant3 1043	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> (♯ ` <" X Y "> ) = 2 )
5		s2fv0g 11367	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( <" X Y "> ` 0 ) = X )
6		s2fv1g 11368	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( <" X Y "> ` 1 ) = Y )
7	5, 6	preq12d 3756	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } = { X , Y } )
8	7	eqcomd 2237	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> { X , Y } = { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } )
9	8	eleq1d 2300	. . . 4 |- ( ( X e. V /\ Y e. V ) -> ( { X , Y } e. E <-> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E ) )
10	9	biimp3a 1381	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E )
11	5	3adant3 1043	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> ( <" X Y "> ` 0 ) = X )
12	4, 10, 11	3jca 1203	. 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 5648	. . . 4 |- ( w = <" X Y "> -> ( (♯ ` w ) = 2 <-> (♯ ` <" X Y "> ) = 2 ) )
14		fveq1 5638	. . . . . 6 |- ( w = <" X Y "> -> ( w ` 0 ) = ( <" X Y "> ` 0 ) )
15		fveq1 5638	. . . . . 6 |- ( w = <" X Y "> -> ( w ` 1 ) = ( <" X Y "> ` 1 ) )
16	14, 15	preq12d 3756	. . . . 5 |- ( w = <" X Y "> -> { ( w ` 0 ) , ( w ` 1 ) } = { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } )
17	16	eleq1d 2300	. . . 4 |- ( w = <" X Y "> -> ( { ( w ` 0 ) , ( w ` 1 ) } e. E <-> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E ) )
18	14	eqeq1d 2240	. . . 4 |- ( w = <" X Y "> -> ( ( w ` 0 ) = X <-> ( <" X Y "> ` 0 ) = X ) )
19	13, 17, 18	3anbi123d 1348	. . 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 16285	. . 3 |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }
24	19, 23	elrab2 2965	. 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 1004   = wceq 1397   e. wcel 2202   {cpr 3670   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   2c2 9193  ♯chash 11036  Word cword 11112   <"cs2 11329  Vtxcvtx 15862  Edgcedg 15907  ClWWalksNOncclwwlknon 16276

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-lsw 11158  df-concat 11167  df-s1 11192  df-s2 11336  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864  df-clwwlk 16242  df-clwwlkn 16254  df-clwwlknon 16277

This theorem is referenced by: (None)