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Theorem 0pth 23420
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 4417 . . 3  |-  (/)  e.  _V
2 ispth 23418 . . 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 683 . 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 969 . . . 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 5470 . . . . . . 7  |-  Fun  (/)
7 cnv0 5235 . . . . . . . 8  |-  `' (/)  =  (/)
87funeqi 5433 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
96, 8mpbir 209 . . . . . 6  |-  Fun  `' (/)
10 hash0 12127 . . . . . . . . . . . 12  |-  ( # `  (/) )  =  0
11 0le1 9855 . . . . . . . . . . . 12  |-  0  <_  1
1210, 11eqbrtri 4306 . . . . . . . . . . 11  |-  ( # `  (/) )  <_  1
13 1z 10668 . . . . . . . . . . . 12  |-  1  e.  ZZ
14 0z 10649 . . . . . . . . . . . . 13  |-  0  e.  ZZ
1510, 14eqeltri 2508 . . . . . . . . . . . 12  |-  ( # `  (/) )  e.  ZZ
16 fzon 11563 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  ( # `  (/) )  e.  ZZ )  ->  (
( # `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) ) )
1713, 15, 16mp2an 672 . . . . . . . . . . 11  |-  ( (
# `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) )
1812, 17mpbi 208 . . . . . . . . . 10  |-  ( 1..^ ( # `  (/) ) )  =  (/)
1918reseq2i 5102 . . . . . . . . 9  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  ( P  |`  (/) )
20 res0 5110 . . . . . . . . 9  |-  ( P  |`  (/) )  =  (/)
2119, 20eqtri 2458 . . . . . . . 8  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  (/)
2221cnveqi 5009 . . . . . . 7  |-  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  `' (/)
2322funeqi 5433 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  <->  Fun  `' (/) )
249, 23mpbir 209 . . . . 5  |-  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )
2518imaeq2i 5162 . . . . . . . 8  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  ( P
" (/) )
26 ima0 5179 . . . . . . . 8  |-  ( P
" (/) )  =  (/)
2725, 26eqtri 2458 . . . . . . 7  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  (/)
2827ineq2i 3544 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  ( ( P " {
0 ,  ( # `  (/) ) } )  i^i  (/) )
29 in0 3658 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  (/) )  =  (/)
3028, 29eqtri 2458 . . . . 5  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/)
3124, 30pm3.2i 455 . . . 4  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) )
3231biantru 505 . . 3  |-  ( (/) ( V Trails  E ) P  <-> 
( (/) ( V Trails  E
) P  /\  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) ) ) )
335, 32syl6bbr 263 . 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 23396 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
353, 33, 343bitrd 279 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  P : ( 0 ... 0 ) --> V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322   (/)c0 3632   {cpr 3874   class class class wbr 4287   `'ccnv 4834    |` cres 4837   "cima 4838   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275    <_ cle 9411   ZZcz 10638   ...cfz 11429  ..^cfzo 11540   #chash 12095   Trails ctrail 23357   Paths cpath 23358
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-wlk 23366  df-trail 23367  df-pth 23368
This theorem is referenced by:  0pthon  23429  0cycl  23464
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