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Theorem constr2pth 23672
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.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
Hypotheses
Ref Expression
2trlY.i  |-  ( I  e.  U  /\  J  e.  W )
2trlY.f  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
2trlY.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
constr2pth  |-  ( ( ( 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 } )  ->  F ( V Paths  E
) P ) )

Proof of Theorem constr2pth
StepHypRef Expression
1 2trlY.i . . . . . 6  |-  ( I  e.  U  /\  J  e.  W )
2 2trlY.f . . . . . 6  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
3 2trlY.p . . . . . 6  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
41, 2, 3constr2trl 23670 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( I  =/= 
J  /\  ( E `  I )  =  { A ,  B }  /\  ( E `  J
)  =  { B ,  C } )  ->  F ( V Trails  E
) P ) )
543adant3 1008 . . . 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 } )  ->  F ( V Trails  E
) P ) )
65imp 429 . . 3  |-  ( ( ( ( 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 } ) )  ->  F ( V Trails  E
) P )
732pthlem1 23666 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  Fun  `' ( P  |`  ( 1..^ 2 ) ) )
81, 22trllemA 23621 . . . . . . 7  |-  ( # `  F )  =  2
9 oveq2 6211 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  (
1..^ ( # `  F
) )  =  ( 1..^ 2 ) )
109reseq2d 5221 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  ( P  |`  ( 1..^ (
# `  F )
) )  =  ( P  |`  ( 1..^ 2 ) ) )
1110cnveqd 5126 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  `' ( P  |`  ( 1..^ ( # `  F
) ) )  =  `' ( P  |`  ( 1..^ 2 ) ) )
1211funeqd 5550 . . . . . . 7  |-  ( (
# `  F )  =  2  ->  ( Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  <->  Fun  `' ( P  |`  ( 1..^ 2 ) ) ) )
138, 12ax-mp 5 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  <->  Fun  `' ( P  |`  ( 1..^ 2 ) ) )
147, 13sylibr 212 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) ) )
15143ad2ant2 1010 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
1615adantr 465 . . 3  |-  ( ( ( ( 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 } ) )  ->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
1732pthlem2 23667 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( ( P " { 0 ,  2 } )  i^i  ( P " ( 1..^ 2 ) ) )  =  (/) )
18 preq2 4066 . . . . . . . . . . 11  |-  ( (
# `  F )  =  2  ->  { 0 ,  ( # `  F
) }  =  {
0 ,  2 } )
1918imaeq2d 5280 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  ( P " { 0 ,  ( # `  F
) } )  =  ( P " {
0 ,  2 } ) )
209imaeq2d 5280 . . . . . . . . . 10  |-  ( (
# `  F )  =  2  ->  ( P " ( 1..^ (
# `  F )
) )  =  ( P " ( 1..^ 2 ) ) )
2119, 20ineq12d 3664 . . . . . . . . 9  |-  ( (
# `  F )  =  2  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  ( ( P " { 0 ,  2 } )  i^i  ( P "
( 1..^ 2 ) ) ) )
2221eqeq1d 2456 . . . . . . . 8  |-  ( (
# `  F )  =  2  ->  (
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) 
<->  ( ( P " { 0 ,  2 } )  i^i  ( P " ( 1..^ 2 ) ) )  =  (/) ) )
238, 22ax-mp 5 . . . . . . 7  |-  ( ( ( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/)  <->  ( ( P " { 0 ,  2 } )  i^i  ( P " (
1..^ 2 ) ) )  =  (/) )
2417, 23sylibr 212 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
25243adantr2 1148 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/) )
26253adant1 1006 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
2726adantr 465 . . 3  |-  ( ( ( ( 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 } ) )  -> 
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )
28 prex 4645 . . . . . . . . 9  |-  { <. 0 ,  I >. , 
<. 1 ,  J >. }  e.  _V
292, 28eqeltri 2538 . . . . . . . 8  |-  F  e. 
_V
30 tpex 6492 . . . . . . . . 9  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
313, 30eqeltri 2538 . . . . . . . 8  |-  P  e. 
_V
3229, 31pm3.2i 455 . . . . . . 7  |-  ( F  e.  _V  /\  P  e.  _V )
3332jctr 542 . . . . . 6  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
34333ad2ant1 1009 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3534adantr 465 . . . 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 } ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
36 ispth 23639 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  _V  /\  P  e. 
_V ) )  -> 
( F ( V Paths 
E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
3735, 36syl 16 . . 3  |-  ( ( ( ( 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 } ) )  -> 
( F ( V Paths 
E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
386, 16, 27, 37mpbir3and 1171 . 2  |-  ( ( ( ( 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 } ) )  ->  F ( V Paths  E
) P )
3938ex 434 1  |-  ( ( ( 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 } )  ->  F ( V Paths  E
) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078    i^i cin 3438   (/)c0 3748   {cpr 3990   {ctp 3992   <.cop 3994   class class class wbr 4403   `'ccnv 4950    |` cres 4953   "cima 4954   Fun wfun 5523   ` cfv 5529  (class class class)co 6203   0cc0 9396   1c1 9397   2c2 10485  ..^cfzo 11668   #chash 12223   Trails ctrail 23578   Paths cpath 23579
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-wlk 23587  df-trail 23588  df-pth 23589
This theorem is referenced by:  2pthon  23673
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