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Theorem seqval 11289
Description: Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
seqval.1  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
Assertion
Ref Expression
seqval  |-  seq  M
(  .+  ,  F
)  =  ran  R
Distinct variable groups:    w, F, x, y, z    w,  .+ , x, y, z    x, M, y
Allowed substitution hints:    R( x, y, z, w)    M( z, w)

Proof of Theorem seqval
StepHypRef Expression
1 df-ima 4850 . 2  |-  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )  =  ran  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )  |`  om )
2 df-seq 11279 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
3 seqval.1 . . . 4  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
4 eqid 2404 . . . . . . 7  |-  _V  =  _V
5 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
6 vex 2919 . . . . . . . . 9  |-  y  e. 
_V
7 oveq1 6047 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
z  +  1 )  =  ( x  + 
1 ) )
87fveq2d 5691 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
98oveq2d 6056 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( x  +  1 ) ) ) )
10 oveq1 6047 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
11 eqid 2404 . . . . . . . . . 10  |-  ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
12 ovex 6065 . . . . . . . . . 10  |-  ( y 
.+  ( F `  ( x  +  1
) ) )  e. 
_V
139, 10, 11, 12ovmpt2 6168 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
145, 6, 13mp2an 654 . . . . . . . 8  |-  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) )
1514opeq2i 3948 . . . . . . 7  |-  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
164, 4, 15mpt2eq123i 6096 . . . . . 6  |-  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >. )  =  ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )
17 rdgeq1 6628 . . . . . 6  |-  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
)  =  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  ->  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x ( z  e. 
_V ,  w  e. 
_V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )
>. ) ,  <. M , 
( F `  M
) >. )  =  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) )
1816, 17ax-mp 8 . . . . 5  |-  rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )  =  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
1918reseq1i 5101 . . . 4  |-  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x ( z  e. 
_V ,  w  e. 
_V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )
>. ) ,  <. M , 
( F `  M
) >. )  |`  om )  =  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
203, 19eqtri 2424 . . 3  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
2120rneqi 5055 . 2  |-  ran  R  =  ran  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
221, 2, 213eqtr4i 2434 1  |-  seq  M
(  .+  ,  F
)  =  ran  R
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   omcom 4804   ran crn 4838    |` cres 4839   "cima 4840   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   reccrdg 6626   1c1 8947    + caddc 8949    seq cseq 11278
This theorem is referenced by:  seqfn  11290  seq1  11291  seqp1  11293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-seq 11279
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