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Theorem el2wlkonotot1 24697
Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
Assertion
Ref Expression
el2wlkonotot1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <->  ( A  =  R  /\  C  =  S  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )

Proof of Theorem el2wlkonotot1
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el2wlkonotot0 24695 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <->  ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
2 simpll 753 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  ( A  =  R  /\  C  =  S ) )  -> 
( V  e.  X  /\  E  e.  Y
) )
3 eleq1 2539 . . . . . . . . . . 11  |-  ( R  =  A  ->  ( R  e.  V  <->  A  e.  V ) )
43eqcoms 2479 . . . . . . . . . 10  |-  ( A  =  R  ->  ( R  e.  V  <->  A  e.  V ) )
5 eleq1 2539 . . . . . . . . . . 11  |-  ( S  =  C  ->  ( S  e.  V  <->  C  e.  V ) )
65eqcoms 2479 . . . . . . . . . 10  |-  ( C  =  S  ->  ( S  e.  V  <->  C  e.  V ) )
74, 6bi2anan9 871 . . . . . . . . 9  |-  ( ( A  =  R  /\  C  =  S )  ->  ( ( R  e.  V  /\  S  e.  V )  <->  ( A  e.  V  /\  C  e.  V ) ) )
87biimpcd 224 . . . . . . . 8  |-  ( ( R  e.  V  /\  S  e.  V )  ->  ( ( A  =  R  /\  C  =  S )  ->  ( A  e.  V  /\  C  e.  V )
) )
98adantl 466 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( ( A  =  R  /\  C  =  S )  ->  ( A  e.  V  /\  C  e.  V )
) )
109imp 429 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  ( A  =  R  /\  C  =  S ) )  -> 
( A  e.  V  /\  C  e.  V
) )
11 el2wlkonotot 24696 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
122, 10, 11syl2anc 661 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  ( A  =  R  /\  C  =  S ) )  -> 
( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
1312bicomd 201 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  ( A  =  R  /\  C  =  S ) )  -> 
( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  <->  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) )
1413pm5.32da 641 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( ( ( A  =  R  /\  C  =  S )  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  <-> 
( ( A  =  R  /\  C  =  S )  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
15 df-3an 975 . . 3  |-  ( ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  <->  ( ( A  =  R  /\  C  =  S )  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
16 df-3an 975 . . 3  |-  ( ( A  =  R  /\  C  =  S  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) )  <->  ( ( A  =  R  /\  C  =  S )  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) )
1714, 15, 163bitr4g 288 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  ( A  =  R  /\  C  =  S  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
181, 17bitrd 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <->  ( A  =  R  /\  C  =  S  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   <.cotp 4041   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505   2c2 10597   #chash 12385   Walks cwalk 24321   2WalksOnOt c2wlkonot 24678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-word 12523  df-wlk 24331  df-wlkon 24337  df-2wlkonot 24681
This theorem is referenced by:  usg2spthonot0  24712
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