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Theorem iscycl 23514
Description: Properties of a pair of functions to be a cycle (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Assertion
Ref Expression
iscycl  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Cycles  E ) P  <->  ( F
( V Paths  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )

Proof of Theorem iscycl
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cycls 23512 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Cycles  E )  =  { <. f ,  p >.  |  (
f ( V Paths  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
2 fveq1 5693 . . . . 5  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
32adantl 466 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  0
)  =  ( P `
 0 ) )
4 simpr 461 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
5 fveq2 5694 . . . . . 6  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
65adantr 465 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
74, 6fveq12d 5700 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  ( # `
 f ) )  =  ( P `  ( # `  F ) ) )
83, 7eqeq12d 2457 . . 3  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p ` 
0 )  =  ( p `  ( # `  f ) )  <->  ( P `  0 )  =  ( P `  ( # `
 F ) ) ) )
91, 8isprmpt2 6746 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( F  e.  W  /\  P  e.  Z )  ->  ( F ( V Cycles  E
) P  <->  ( F
( V Paths  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) ) )
109imp 429 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Cycles  E ) P  <->  ( F
( V Paths  E ) P  /\  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   0cc0 9285   #chash 12106   Paths cpath 23410   Cycles ccycl 23417
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 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-n0 10583  df-z 10650  df-uz 10865  df-fz 11441  df-fzo 11552  df-word 12232  df-wlk 23418  df-trail 23419  df-pth 23420  df-cycl 23423
This theorem is referenced by:  0cycl  23516  cyclispth  23518  cycliscrct  23519  cyclnspth  23520  cyclispthon  23522  usgrcyclnl1  23529  usgrcyclnl2  23530  3v3e3cycl1  23533  constr3cycl  23550  4cycl4v4e  23555  4cycl4dv4e  23557
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