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Theorem 2wlkonotv 30531
Description: If an ordered tripple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
Assertion
Ref Expression
2wlkonotv  |-  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) ) )

Proof of Theorem 2wlkonotv
StepHypRef Expression
1 2wlkonot3v 30529 . 2  |-  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  <. A ,  B ,  C >.  e.  ( ( V  X.  V )  X.  V
) ) )
2 3simpb 986 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  <. A ,  B ,  C >.  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  <. A ,  B ,  C >.  e.  (
( V  X.  V
)  X.  V ) ) )
3 df-ot 3981 . . . . 5  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
43eleq1i 2526 . . . 4  |-  ( <. A ,  B ,  C >.  e.  ( ( V  X.  V )  X.  V )  <->  <. <. A ,  B >. ,  C >.  e.  ( ( V  X.  V )  X.  V
) )
5 opelxp 4964 . . . . 5  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( V  X.  V
)  X.  V )  <-> 
( <. A ,  B >.  e.  ( V  X.  V )  /\  C  e.  V ) )
6 opelxp 4964 . . . . . 6  |-  ( <. A ,  B >.  e.  ( V  X.  V
)  <->  ( A  e.  V  /\  B  e.  V ) )
7 df-3an 967 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  <->  ( ( A  e.  V  /\  B  e.  V
)  /\  C  e.  V ) )
87biimpri 206 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  C  e.  V )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)
96, 8sylanb 472 . . . . 5  |-  ( (
<. A ,  B >.  e.  ( V  X.  V
)  /\  C  e.  V )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)
105, 9sylbi 195 . . . 4  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( V  X.  V
)  X.  V )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
114, 10sylbi 195 . . 3  |-  ( <. A ,  B ,  C >.  e.  ( ( V  X.  V )  X.  V )  -> 
( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )
1211anim2i 569 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  <. A ,  B ,  C >.  e.  (
( V  X.  V
)  X.  V ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )
131, 2, 123syl 20 1  |-  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758   _Vcvv 3065   <.cop 3978   <.cotp 3980    X. cxp 4933  (class class class)co 6187   2WalksOnOt c2wlkonot 30509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-ot 3981  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-2wlkonot 30512
This theorem is referenced by:  frg2woteqm  30787
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