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Theorem iscrct 25197
Description: Properties of a pair of functions to be a circuit (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Assertion
Ref Expression
iscrct  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Circuits  E ) P  <->  ( F
( V Trails  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )

Proof of Theorem iscrct
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crcts 25195 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Circuits  E )  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )
2 fveq1 5880 . . . . 5  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
32adantl 467 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  0
)  =  ( P `
 0 ) )
4 simpr 462 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
5 fveq2 5881 . . . . . 6  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
65adantr 466 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
74, 6fveq12d 5887 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  ( # `
 f ) )  =  ( P `  ( # `  F ) ) )
83, 7eqeq12d 2451 . . 3  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p ` 
0 )  =  ( p `  ( # `  f ) )  <->  ( P `  0 )  =  ( P `  ( # `
 F ) ) ) )
91, 8isprmpt2 6979 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( F  e.  W  /\  P  e.  Z )  ->  ( F ( V Circuits  E ) P  <->  ( F ( V Trails  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) ) )
109imp 430 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Circuits  E ) P  <->  ( F
( V Trails  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   0cc0 9538   #chash 12512   Trails ctrail 25072   Circuits ccrct 25079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-wlk 25081  df-trail 25082  df-crct 25085
This theorem is referenced by:  0crct  25199  crctistrl  25201  cycliscrct  25203
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