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Theorem 2wlkonotv 25079
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 25077 . 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 992 . 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 4025 . . . . 5  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
43eleq1i 2531 . . . 4  |-  ( <. A ,  B ,  C >.  e.  ( ( V  X.  V )  X.  V )  <->  <. <. A ,  B >. ,  C >.  e.  ( ( V  X.  V )  X.  V
) )
5 opelxp 5018 . . . . 5  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( V  X.  V
)  X.  V )  <-> 
( <. A ,  B >.  e.  ( V  X.  V )  /\  C  e.  V ) )
6 opelxp 5018 . . . . . 6  |-  ( <. A ,  B >.  e.  ( V  X.  V
)  <->  ( A  e.  V  /\  B  e.  V ) )
7 df-3an 973 . . . . . . 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 470 . . . . 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 567 . 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 367    /\ w3a 971    e. wcel 1823   _Vcvv 3106   <.cop 4022   <.cotp 4024    X. cxp 4986  (class class class)co 6270   2WalksOnOt c2wlkonot 25057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-ot 4025  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-2wlkonot 25060
This theorem is referenced by:  frg2woteqm  25261
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