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Theorem 0pth 24550
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 4567 . . 3  |-  (/)  e.  _V
2 ispth 24548 . . 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 978 . . . 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 5635 . . . . . . 7  |-  Fun  (/)
7 cnv0 5399 . . . . . . . 8  |-  `' (/)  =  (/)
87funeqi 5598 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
96, 8mpbir 209 . . . . . 6  |-  Fun  `' (/)
10 hash0 12419 . . . . . . . . . . . 12  |-  ( # `  (/) )  =  0
11 0le1 10083 . . . . . . . . . . . 12  |-  0  <_  1
1210, 11eqbrtri 4456 . . . . . . . . . . 11  |-  ( # `  (/) )  <_  1
13 1z 10901 . . . . . . . . . . . 12  |-  1  e.  ZZ
14 0z 10882 . . . . . . . . . . . . 13  |-  0  e.  ZZ
1510, 14eqeltri 2527 . . . . . . . . . . . 12  |-  ( # `  (/) )  e.  ZZ
16 fzon 11829 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  ( # `  (/) )  e.  ZZ )  ->  (
( # `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) ) )
1713, 15, 16mp2an 672 . . . . . . . . . . 11  |-  ( (
# `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) )
1812, 17mpbi 208 . . . . . . . . . 10  |-  ( 1..^ ( # `  (/) ) )  =  (/)
1918reseq2i 5260 . . . . . . . . 9  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  ( P  |`  (/) )
20 res0 5268 . . . . . . . . 9  |-  ( P  |`  (/) )  =  (/)
2119, 20eqtri 2472 . . . . . . . 8  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  (/)
2221cnveqi 5167 . . . . . . 7  |-  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  `' (/)
2322funeqi 5598 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  <->  Fun  `' (/) )
249, 23mpbir 209 . . . . 5  |-  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )
2518imaeq2i 5325 . . . . . . . 8  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  ( P
" (/) )
26 ima0 5342 . . . . . . . 8  |-  ( P
" (/) )  =  (/)
2725, 26eqtri 2472 . . . . . . 7  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  (/)
2827ineq2i 3682 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  ( ( P " {
0 ,  ( # `  (/) ) } )  i^i  (/) )
29 in0 3797 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  (/) )  =  (/)
3028, 29eqtri 2472 . . . . 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 24526 . 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 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    i^i cin 3460   (/)c0 3770   {cpr 4016   class class class wbr 4437   `'ccnv 4988    |` cres 4991   "cima 4992   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   0cc0 9495   1c1 9496    <_ cle 9632   ZZcz 10871   ...cfz 11683  ..^cfzo 11806   #chash 12387   Trails ctrail 24477   Paths cpath 24478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-wlk 24486  df-trail 24487  df-pth 24488
This theorem is referenced by:  0pthon  24559  0cycl  24605
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