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Theorem 2spotfi 25612
Description: In a finite graph, the set of simple paths of length 2 between two vertices (as ordered triples) is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
Assertion
Ref Expression
2spotfi  |-  ( ( ( V  e.  Fin  /\  E  e.  X )  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) B )  e.  Fin )

Proof of Theorem 2spotfi
Dummy variables  f  p  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2spthonot 25586 . 2  |-  ( ( ( V  e.  Fin  /\  E  e.  X )  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) } )
2 3xpfi 7847 . . . 4  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
3 rabfi 7800 . . . 4  |-  ( ( ( V  X.  V
)  X.  V )  e.  Fin  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) }  e.  Fin )
42, 3syl 17 . . 3  |-  ( V  e.  Fin  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) }  e.  Fin )
54ad2antrr 731 . 2  |-  ( ( ( V  e.  Fin  /\  E  e.  X )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) B ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  B ) ) }  e.  Fin )
61, 5eqeltrd 2511 1  |-  ( ( ( V  e.  Fin  /\  E  e.  X )  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) B )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438   E.wex 1660    e. wcel 1869   {crab 2780   class class class wbr 4421    X. cxp 4849   ` cfv 5599  (class class class)co 6303   1stc1st 6803   2ndc2nd 6804   Fincfn 7575   1c1 9542   2c2 10661   #chash 12516   SPathOn cspthon 25225   2SPathOnOt c2pthonot 25577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-fin 7579  df-2spthonot 25580
This theorem is referenced by:  frghash2spot  25783
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