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Theorem wlkiswwlk2lem2 24813
Description: Lemma 2 for wlkiswwlk2 24818. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
Hypothesis
Ref Expression
wlkiswwlk2lem.f  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } ) )
Assertion
Ref Expression
wlkiswwlk2lem2  |-  ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  ->  ( F `  I )  =  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } ) )
Distinct variable groups:    x, E    x, I    x, P
Allowed substitution hint:    F( x)

Proof of Theorem wlkiswwlk2lem2
StepHypRef Expression
1 wlkiswwlk2lem.f . . 3  |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) )  |->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } ) )
21a1i 11 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  ->  F  =  ( x  e.  ( 0..^ ( (
# `  P )  -  1 ) ) 
|->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) } ) ) )
3 fveq2 5774 . . . . 5  |-  ( x  =  I  ->  ( P `  x )  =  ( P `  I ) )
4 oveq1 6203 . . . . . 6  |-  ( x  =  I  ->  (
x  +  1 )  =  ( I  + 
1 ) )
54fveq2d 5778 . . . . 5  |-  ( x  =  I  ->  ( P `  ( x  +  1 ) )  =  ( P `  ( I  +  1
) ) )
63, 5preq12d 4031 . . . 4  |-  ( x  =  I  ->  { ( P `  x ) ,  ( P `  ( x  +  1
) ) }  =  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } )
76fveq2d 5778 . . 3  |-  ( x  =  I  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  =  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } ) )
87adantl 464 . 2  |-  ( ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  /\  x  =  I )  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  =  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } ) )
9 simpr 459 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  ->  I  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )
10 fvex 5784 . . 3  |-  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1
) ) } )  e.  _V
1110a1i 11 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  ->  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1
) ) } )  e.  _V )
122, 8, 9, 11fvmptd 5862 1  |-  ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  ->  ( F `  I )  =  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   {cpr 3946    |-> cmpt 4425   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406    - cmin 9718   NN0cn0 10712  ..^cfzo 11717   #chash 12307
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-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2737  df-rex 2738  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-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199
This theorem is referenced by:  wlkiswwlk2lem4  24815
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