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Theorem 2wlklem 24983
Description: Lemma for is2wlk 24984 and 2wlklemA 24973. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Assertion
Ref Expression
2wlklem  |-  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
Distinct variable groups:    k, E    k, F    P, k

Proof of Theorem 2wlklem
StepHypRef Expression
1 c0ex 9620 . 2  |-  0  e.  _V
2 1ex 9621 . 2  |-  1  e.  _V
3 fveq2 5849 . . . 4  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
43fveq2d 5853 . . 3  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
5 fveq2 5849 . . . 4  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
6 oveq1 6285 . . . . . 6  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
7 0p1e1 10688 . . . . . 6  |-  ( 0  +  1 )  =  1
86, 7syl6eq 2459 . . . . 5  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
98fveq2d 5853 . . . 4  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
105, 9preq12d 4059 . . 3  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
114, 10eqeq12d 2424 . 2  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
12 fveq2 5849 . . . 4  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
1312fveq2d 5853 . . 3  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
14 fveq2 5849 . . . 4  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
15 oveq1 6285 . . . . . 6  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
16 1p1e2 10690 . . . . . 6  |-  ( 1  +  1 )  =  2
1715, 16syl6eq 2459 . . . . 5  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
1817fveq2d 5853 . . . 4  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
1914, 18preq12d 4059 . . 3  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
2013, 19eqeq12d 2424 . 2  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
211, 2, 11, 20ralpr 4025 1  |-  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405   A.wral 2754   {cpr 3974   ` cfv 5569  (class class class)co 6278   0cc0 9522   1c1 9523    + caddc 9525   2c2 10626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-2 10635
This theorem is referenced by:  is2wlk  24984  2wlklem1  25016  usgra2wlkspthlem1  25036  usgra2wlkspthlem2  25037  usgrcyclnl2  25058  4cycl4v4e  25083  4cycl4dv4e  25085  usgra2pthspth  37980  usgra2pthlem1  37982  usgra2pth  37983
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