Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0pth Unicode version

Theorem 0pth 28356
Description: A pair of an empty set (of edges) and a second set (of vertices) is a path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Assertion
Ref Expression
0pth  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  P : ( 0 ... 0 ) --> V ) )

Proof of Theorem 0pth
StepHypRef Expression
1 0ex 4166 . . 3  |-  (/)  e.  _V
2 ispth 28354 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( (/)  e.  _V  /\  P  e.  Z ) )  ->  ( (/) ( V Paths 
E ) P  <->  ( (/) ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) )
31, 2mpanr1 664 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  ( (/) ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) )
4 3anass 938 . . . 4  |-  ( (
(/) ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) 
<->  ( (/) ( V Trails  E
) P  /\  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) )
54a1i 10 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  (
( (/) ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) 
<->  ( (/) ( V Trails  E
) P  /\  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) ) )
6 fun0 5323 . . . . . . 7  |-  Fun  (/)
7 cnv0 5100 . . . . . . . 8  |-  `' (/)  =  (/)
87funeqi 5291 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
96, 8mpbir 200 . . . . . 6  |-  Fun  `' (/)
10 hash0 11371 . . . . . . . . . . . 12  |-  ( # `  (/) )  =  0
11 0le1 9313 . . . . . . . . . . . 12  |-  0  <_  1
1210, 11eqbrtri 4058 . . . . . . . . . . 11  |-  ( # `  (/) )  <_  1
13 1z 10069 . . . . . . . . . . . 12  |-  1  e.  ZZ
14 0z 10051 . . . . . . . . . . . . 13  |-  0  e.  ZZ
1510, 14eqeltri 2366 . . . . . . . . . . . 12  |-  ( # `  (/) )  e.  ZZ
16 fzon 28212 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  ( # `  (/) )  e.  ZZ )  ->  (
( # `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) ) )
1713, 15, 16mp2an 653 . . . . . . . . . . 11  |-  ( (
# `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) )
1812, 17mpbi 199 . . . . . . . . . 10  |-  ( 1..^ ( # `  (/) ) )  =  (/)
1918reseq2i 4968 . . . . . . . . 9  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  ( P  |`  (/) )
20 res0 4975 . . . . . . . . 9  |-  ( P  |`  (/) )  =  (/)
2119, 20eqtri 2316 . . . . . . . 8  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  (/)
2221cnveqi 4872 . . . . . . 7  |-  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  `' (/)
2322funeqi 5291 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  <->  Fun  `' (/) )
249, 23mpbir 200 . . . . 5  |-  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )
2518imaeq2i 5026 . . . . . . . 8  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  ( P
" (/) )
26 ima0 5046 . . . . . . . 8  |-  ( P
" (/) )  =  (/)
2725, 26eqtri 2316 . . . . . . 7  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  (/)
2827ineq2i 3380 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  ( ( P " {
0 ,  ( # `  (/) ) } )  i^i  (/) )
29 in0 3493 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  (/) )  =  (/)
3028, 29eqtri 2316 . . . . 5  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/)
3124, 30pm3.2i 441 . . . 4  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) )
3231biantru 491 . . 3  |-  ( (/) ( V Trails  E ) P  <-> 
( (/) ( V Trails  E
) P  /\  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) )
335, 32syl6bbr 254 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  (
( (/) ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) 
<->  (/) ( V Trails  E ) P ) )
34 0trl 28344 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
353, 33, 343bitrd 270 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   (/)c0 3468   {cpr 3654   class class class wbr 4039   `'ccnv 4704    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    <_ cle 8884   ZZcz 10040   ...cfz 10798  ..^cfzo 10886   #chash 11353   Trails ctrail 28311   Paths cpath 28312
This theorem is referenced by:  0cycl  28372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-wlk 28319  df-trail 28320  df-pth 28321
  Copyright terms: Public domain W3C validator