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Theorem 2spotfi 30551
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 30525 . 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 xpfi 7686 . . . . . 6  |-  ( ( V  e.  Fin  /\  V  e.  Fin )  ->  ( V  X.  V
)  e.  Fin )
32anidms 645 . . . . 5  |-  ( V  e.  Fin  ->  ( V  X.  V )  e. 
Fin )
4 xpfi 7686 . . . . 5  |-  ( ( ( V  X.  V
)  e.  Fin  /\  V  e.  Fin )  ->  ( ( V  X.  V )  X.  V
)  e.  Fin )
53, 4mpancom 669 . . . 4  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
6 rabfi 7640 . . . 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 )
75, 6syl 16 . . 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 )
87ad2antrr 725 . 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 )
91, 8eqeltrd 2539 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 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {crab 2799   class class class wbr 4392    X. cxp 4938   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   Fincfn 7412   1c1 9386   2c2 10474   #chash 12206   SPathOn cspthon 23549   2SPathOnOt c2pthonot 30516
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-fin 7416  df-2spthonot 30519
This theorem is referenced by:  frghash2spot  30796
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