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Theorem 0pth 21570
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 4339 . . 3  |-  (/)  e.  _V
2 ispth 21568 . . 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 665 . 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 940 . . . 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 11 . . 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 5508 . . . . . . 7  |-  Fun  (/)
7 cnv0 5275 . . . . . . . 8  |-  `' (/)  =  (/)
87funeqi 5474 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
96, 8mpbir 201 . . . . . 6  |-  Fun  `' (/)
10 hash0 11646 . . . . . . . . . . . 12  |-  ( # `  (/) )  =  0
11 0le1 9551 . . . . . . . . . . . 12  |-  0  <_  1
1210, 11eqbrtri 4231 . . . . . . . . . . 11  |-  ( # `  (/) )  <_  1
13 1z 10311 . . . . . . . . . . . 12  |-  1  e.  ZZ
14 0z 10293 . . . . . . . . . . . . 13  |-  0  e.  ZZ
1510, 14eqeltri 2506 . . . . . . . . . . . 12  |-  ( # `  (/) )  e.  ZZ
16 fzon 11158 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  ( # `  (/) )  e.  ZZ )  ->  (
( # `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) ) )
1713, 15, 16mp2an 654 . . . . . . . . . . 11  |-  ( (
# `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) )
1812, 17mpbi 200 . . . . . . . . . 10  |-  ( 1..^ ( # `  (/) ) )  =  (/)
1918reseq2i 5143 . . . . . . . . 9  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  ( P  |`  (/) )
20 res0 5150 . . . . . . . . 9  |-  ( P  |`  (/) )  =  (/)
2119, 20eqtri 2456 . . . . . . . 8  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  (/)
2221cnveqi 5047 . . . . . . 7  |-  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  `' (/)
2322funeqi 5474 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  <->  Fun  `' (/) )
249, 23mpbir 201 . . . . 5  |-  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )
2518imaeq2i 5201 . . . . . . . 8  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  ( P
" (/) )
26 ima0 5221 . . . . . . . 8  |-  ( P
" (/) )  =  (/)
2725, 26eqtri 2456 . . . . . . 7  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  (/)
2827ineq2i 3539 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  ( ( P " {
0 ,  ( # `  (/) ) } )  i^i  (/) )
29 in0 3653 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  (/) )  =  (/)
3028, 29eqtri 2456 . . . . 5  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/)
3124, 30pm3.2i 442 . . . 4  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) )
3231biantru 492 . . 3  |-  ( (/) ( V Trails  E ) P  <-> 
( (/) ( V Trails  E
) P  /\  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) )
335, 32syl6bbr 255 . 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 21546 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
353, 33, 343bitrd 271 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   (/)c0 3628   {cpr 3815   class class class wbr 4212   `'ccnv 4877    |` cres 4880   "cima 4881   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    <_ cle 9121   ZZcz 10282   ...cfz 11043  ..^cfzo 11135   #chash 11618   Trails ctrail 21507   Paths cpath 21508
This theorem is referenced by:  0pthon  21579  0cycl  21614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-wlk 21516  df-trail 21517  df-pth 21518
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