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Theorem wlkiswwlk2lem5 24816
Description: Lemma 5 for wlkiswwlk2 24818. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Hypothesis
Ref Expression
wlkiswwlk2lem.f  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } ) )
Assertion
Ref Expression
wlkiswwlk2lem5  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  ->  F  e. Word  dom  E
) )
Distinct variable groups:    i, E, x    i, F    P, i, x    i, V, x
Allowed substitution hint:    F( x)

Proof of Theorem wlkiswwlk2lem5
StepHypRef Expression
1 usgraf1o 24479 . . . . . . 7  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
213ad2ant1 1015 . . . . . 6  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  E : dom  E -1-1-onto-> ran 
E )
32ad2antrr 723 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  E : dom  E -1-1-onto-> ran  E )
4 simpr 459 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) )
5 fveq2 5774 . . . . . . . . . . 11  |-  ( i  =  x  ->  ( P `  i )  =  ( P `  x ) )
6 oveq1 6203 . . . . . . . . . . . 12  |-  ( i  =  x  ->  (
i  +  1 )  =  ( x  + 
1 ) )
76fveq2d 5778 . . . . . . . . . . 11  |-  ( i  =  x  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( x  +  1
) ) )
85, 7preq12d 4031 . . . . . . . . . 10  |-  ( i  =  x  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } )
98eleq1d 2451 . . . . . . . . 9  |-  ( i  =  x  ->  ( { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  x
) ,  ( P `
 ( x  + 
1 ) ) }  e.  ran  E ) )
109adantl 464 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  /\  i  =  x )  ->  ( { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  x
) ,  ( P `
 ( x  + 
1 ) ) }  e.  ran  E ) )
114, 10rspcdv 3138 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  ( A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  ->  { ( P `
 x ) ,  ( P `  (
x  +  1 ) ) }  e.  ran  E ) )
1211impancom 438 . . . . . 6  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  ->  ( x  e.  ( 0..^ ( (
# `  P )  -  1 ) )  ->  { ( P `
 x ) ,  ( P `  (
x  +  1 ) ) }  e.  ran  E ) )
1312imp 427 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  { ( P `  x ) ,  ( P `  ( x  +  1
) ) }  e.  ran  E )
14 f1ocnvdm 6089 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { ( P `
 x ) ,  ( P `  (
x  +  1 ) ) }  e.  ran  E )  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  e.  dom  E )
153, 13, 14syl2anc 659 . . . 4  |-  ( ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  /\  x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  e.  dom  E )
16 wlkiswwlk2lem.f . . . 4  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } ) )
1715, 16fmptd 5957 . . 3  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  ->  F :
( 0..^ ( (
# `  P )  -  1 ) ) --> dom  E )
18 iswrdi 12457 . . 3  |-  ( F : ( 0..^ ( ( # `  P
)  -  1 ) ) --> dom  E  ->  F  e. Word  dom  E )
1917, 18syl 16 . 2  |-  ( ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P
) )  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E )  ->  F  e. Word  dom 
E )
2019ex 432 1  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  ->  F  e. Word  dom  E
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   {cpr 3946   class class class wbr 4367    |-> cmpt 4425   `'ccnv 4912   dom cdm 4913   ran crn 4914   -->wf 5492   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406    <_ cle 9540    - cmin 9718  ..^cfzo 11717   #chash 12307  Word cword 12438   USGrph cusg 24451
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-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-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  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-usgra 24454
This theorem is referenced by:  wlkiswwlk2lem6  24817
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