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Theorem wlkiswwlk2lem2 30475
Description: Lemma 2 for wlkiswwlk2 30480. (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 5800 . . . . 5  |-  ( x  =  I  ->  ( P `  x )  =  ( P `  I ) )
4 oveq1 6208 . . . . . 6  |-  ( x  =  I  ->  (
x  +  1 )  =  ( I  + 
1 ) )
54fveq2d 5804 . . . . 5  |-  ( x  =  I  ->  ( P `  ( x  +  1 ) )  =  ( P `  ( I  +  1
) ) )
63, 5preq12d 4071 . . . 4  |-  ( x  =  I  ->  { ( P `  x ) ,  ( P `  ( x  +  1
) ) }  =  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } )
76fveq2d 5804 . . 3  |-  ( x  =  I  ->  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1
) ) } )  =  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) } ) )
87adantl 466 . 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 461 . 2  |-  ( ( ( # `  P
)  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )  ->  I  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )
10 fvex 5810 . . 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 5889 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   {cpr 3988    |-> cmpt 4459   `'ccnv 4948   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397    - cmin 9707   NN0cn0 10691  ..^cfzo 11666   #chash 12221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204
This theorem is referenced by:  wlkiswwlk2lem4  30477
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