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Theorem 0pth 24693
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 4497 . . 3  |-  (/)  e.  _V
2 ispth 24691 . . 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 681 . 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 975 . . . 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 5553 . . . . . . 7  |-  Fun  (/)
7 cnv0 5319 . . . . . . . 8  |-  `' (/)  =  (/)
87funeqi 5516 . . . . . . 7  |-  ( Fun  `' (/)  <->  Fun  (/) )
96, 8mpbir 209 . . . . . 6  |-  Fun  `' (/)
10 hash0 12340 . . . . . . . . . . . 12  |-  ( # `  (/) )  =  0
11 0le1 9993 . . . . . . . . . . . 12  |-  0  <_  1
1210, 11eqbrtri 4386 . . . . . . . . . . 11  |-  ( # `  (/) )  <_  1
13 1z 10811 . . . . . . . . . . . 12  |-  1  e.  ZZ
14 0z 10792 . . . . . . . . . . . . 13  |-  0  e.  ZZ
1510, 14eqeltri 2466 . . . . . . . . . . . 12  |-  ( # `  (/) )  e.  ZZ
16 fzon 11741 . . . . . . . . . . . 12  |-  ( ( 1  e.  ZZ  /\  ( # `  (/) )  e.  ZZ )  ->  (
( # `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) ) )
1713, 15, 16mp2an 670 . . . . . . . . . . 11  |-  ( (
# `  (/) )  <_ 
1  <->  ( 1..^ (
# `  (/) ) )  =  (/) )
1812, 17mpbi 208 . . . . . . . . . 10  |-  ( 1..^ ( # `  (/) ) )  =  (/)
1918reseq2i 5183 . . . . . . . . 9  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  ( P  |`  (/) )
20 res0 5190 . . . . . . . . 9  |-  ( P  |`  (/) )  =  (/)
2119, 20eqtri 2411 . . . . . . . 8  |-  ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  (/)
2221cnveqi 5090 . . . . . . 7  |-  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  =  `' (/)
2322funeqi 5516 . . . . . 6  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  <->  Fun  `' (/) )
249, 23mpbir 209 . . . . 5  |-  Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )
2518imaeq2i 5247 . . . . . . . 8  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  ( P
" (/) )
26 ima0 5264 . . . . . . . 8  |-  ( P
" (/) )  =  (/)
2725, 26eqtri 2411 . . . . . . 7  |-  ( P
" ( 1..^ (
# `  (/) ) ) )  =  (/)
2827ineq2i 3611 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  ( ( P " {
0 ,  ( # `  (/) ) } )  i^i  (/) )
29 in0 3738 . . . . . 6  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  (/) )  =  (/)
3028, 29eqtri 2411 . . . . 5  |-  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/)
3124, 30pm3.2i 453 . . . 4  |-  ( Fun  `' ( P  |`  ( 1..^ ( # `  (/) ) ) )  /\  ( ( P " { 0 ,  ( # `  (/) ) } )  i^i  ( P
" ( 1..^ (
# `  (/) ) ) ) )  =  (/) )
3231biantru 503 . . 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 24669 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034    i^i cin 3388   (/)c0 3711   {cpr 3946   class class class wbr 4367   `'ccnv 4912    |` cres 4915   "cima 4916   Fun wfun 5490   -->wf 5492   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    <_ cle 9540   ZZcz 10781   ...cfz 11593  ..^cfzo 11717   #chash 12307   Trails ctrail 24620   Paths cpath 24621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-wlk 24629  df-trail 24630  df-pth 24631
This theorem is referenced by:  0pthon  24702  0cycl  24748
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