Theorem s2elclwwlknon2 16591

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 11535	. . 3 |- ( ( X e. V /\ Y e. V ) -> <" X Y "> e. Word V )
2	1	3adant3 1048	. 2 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. Word V )
3		s2leng 11539	. . . 4 |- ( ( X e. V /\ Y e. V ) -> (♯ ` <" X Y "> ) = 2 )
4	3	3adant3 1048	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> (♯ ` <" X Y "> ) = 2 )
5		s2fv0g 11537	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( <" X Y "> ` 0 ) = X )
6		s2fv1g 11538	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( <" X Y "> ` 1 ) = Y )
7	5, 6	preq12d 3792	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } = { X , Y } )
8	7	eqcomd 2244	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> { X , Y } = { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } )
9	8	eleq1d 2307	. . . 4 |- ( ( X e. V /\ Y e. V ) -> ( { X , Y } e. E <-> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E ) )
10	9	biimp3a 1386	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E )
11	5	3adant3 1048	. . 3 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> ( <" X Y "> ` 0 ) = X )
12	4, 10, 11	3jca 1208	. 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 5699	. . . 4 |- ( w = <" X Y "> -> ( (♯ ` w ) = 2 <-> (♯ ` <" X Y "> ) = 2 ) )
14		fveq1 5689	. . . . . 6 |- ( w = <" X Y "> -> ( w ` 0 ) = ( <" X Y "> ` 0 ) )
15		fveq1 5689	. . . . . 6 |- ( w = <" X Y "> -> ( w ` 1 ) = ( <" X Y "> ` 1 ) )
16	14, 15	preq12d 3792	. . . . 5 |- ( w = <" X Y "> -> { ( w ` 0 ) , ( w ` 1 ) } = { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } )
17	16	eleq1d 2307	. . . 4 |- ( w = <" X Y "> -> ( { ( w ` 0 ) , ( w ` 1 ) } e. E <-> { ( <" X Y "> ` 0 ) , ( <" X Y "> ` 1 ) } e. E ) )
18	14	eqeq1d 2247	. . . 4 |- ( w = <" X Y "> -> ( ( w ` 0 ) = X <-> ( <" X Y "> ` 0 ) = X ) )
19	13, 17, 18	3anbi123d 1353	. . 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 16590	. . 3 |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }
24	19, 23	elrab2 2985	. 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 421	1 |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. ( X C 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cpr 3706   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   2c2 9334  ♯chash 11192  Word cword 11282   <"cs2 11499  Vtxcvtx 16167  Edgcedg 16212  ClWWalksNOncclwwlknon 16581

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-lsw 11328  df-concat 11337  df-s1 11362  df-s2 11506  df-ndx 13333  df-slot 13334  df-base 13336  df-vtx 16169  df-clwwlk 16547  df-clwwlkn 16559  df-clwwlknon 16582

This theorem is referenced by: (None)