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Theorem 2pthoncl 23500
Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Assertion
Ref Expression
2pthoncl  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  e.  _V  /\  J  e.  _V )  /\  ( I  =/=  J  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )

Proof of Theorem 2pthoncl
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neeq1 2614 . . . . . . 7  |-  ( i  =  I  ->  (
i  =/=  j  <->  I  =/=  j ) )
2 fveq2 5689 . . . . . . . 8  |-  ( i  =  I  ->  ( E `  i )  =  ( E `  I ) )
32eqeq1d 2449 . . . . . . 7  |-  ( i  =  I  ->  (
( E `  i
)  =  { A ,  B }  <->  ( E `  I )  =  { A ,  B }
) )
4 biidd 237 . . . . . . 7  |-  ( i  =  I  ->  (
( E `  j
)  =  { B ,  C }  <->  ( E `  j )  =  { B ,  C }
) )
51, 3, 43anbi123d 1289 . . . . . 6  |-  ( i  =  I  ->  (
( i  =/=  j  /\  ( E `  i
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  <->  ( I  =/=  j  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
) ) )
6 opeq2 4058 . . . . . . . 8  |-  ( i  =  I  ->  <. 0 ,  i >.  =  <. 0 ,  I >. )
76preq1d 3958 . . . . . . 7  |-  ( i  =  I  ->  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  =  { <. 0 ,  I >. , 
<. 1 ,  j
>. } )
87breq1d 4300 . . . . . 6  |-  ( i  =  I  ->  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  <->  { <. 0 ,  I >. ,  <. 1 ,  j
>. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
95, 8imbi12d 320 . . . . 5  |-  ( i  =  I  ->  (
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )  <->  ( (
I  =/=  j  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  j
>. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
109imbi2d 316 . . . 4  |-  ( i  =  I  ->  (
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  ->  (
( i  =/=  j  /\  ( E `  i
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  ->  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )  <-> 
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  ->  (
( I  =/=  j  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  j
>. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) ) )
11 neeq2 2615 . . . . . . 7  |-  ( j  =  J  ->  (
I  =/=  j  <->  I  =/=  J ) )
12 biidd 237 . . . . . . 7  |-  ( j  =  J  ->  (
( E `  I
)  =  { A ,  B }  <->  ( E `  I )  =  { A ,  B }
) )
13 fveq2 5689 . . . . . . . 8  |-  ( j  =  J  ->  ( E `  j )  =  ( E `  J ) )
1413eqeq1d 2449 . . . . . . 7  |-  ( j  =  J  ->  (
( E `  j
)  =  { B ,  C }  <->  ( E `  J )  =  { B ,  C }
) )
1511, 12, 143anbi123d 1289 . . . . . 6  |-  ( j  =  J  ->  (
( I  =/=  j  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  <->  ( I  =/=  J  /\  ( E `
 I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) ) )
16 opeq2 4058 . . . . . . . 8  |-  ( j  =  J  ->  <. 1 ,  j >.  =  <. 1 ,  J >. )
1716preq2d 3959 . . . . . . 7  |-  ( j  =  J  ->  { <. 0 ,  I >. , 
<. 1 ,  j
>. }  =  { <. 0 ,  I >. , 
<. 1 ,  J >. } )
1817breq1d 4300 . . . . . 6  |-  ( j  =  J  ->  ( { <. 0 ,  I >. ,  <. 1 ,  j
>. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  <->  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } ) )
1915, 18imbi12d 320 . . . . 5  |-  ( j  =  J  ->  (
( ( I  =/=  j  /\  ( E `
 I )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  I >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E
) C ) {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )  <->  ( (
I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
2019imbi2d 316 . . . 4  |-  ( j  =  J  ->  (
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  ->  (
( I  =/=  j  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  j
>. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )  <-> 
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  ->  (
( I  =/=  J  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) ) )
21 2pthon 23499 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
2210, 20, 21vtocl2g 3032 . . 3  |-  ( ( I  e.  _V  /\  J  e.  _V )  ->  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  ->  (
( I  =/=  J  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
2322com12 31 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( I  e. 
_V  /\  J  e.  _V )  ->  ( ( I  =/=  J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } )  ->  { <. 0 ,  I >. , 
<. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
24233imp 1181 1  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
I  e.  _V  /\  J  e.  _V )  /\  ( I  =/=  J  /\  ( E `  I
)  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C } ) )  ->  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970   {cpr 3877   {ctp 3879   <.cop 3881   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281   2c2 10369   PathOn cpthon 23409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-wlk 23413  df-trail 23414  df-pth 23415  df-wlkon 23419  df-pthon 23421
This theorem is referenced by:  2pthon3v  23501
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