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Theorem seqval 11334
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 4891 . 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 11324 . 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 2436 . . . . . . 7  |-  _V  =  _V
5 vex 2959 . . . . . . . . 9  |-  x  e. 
_V
6 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
7 oveq1 6088 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
z  +  1 )  =  ( x  + 
1 ) )
87fveq2d 5732 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
98oveq2d 6097 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( x  +  1 ) ) ) )
10 oveq1 6088 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
11 eqid 2436 . . . . . . . . . 10  |-  ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
12 ovex 6106 . . . . . . . . . 10  |-  ( y 
.+  ( F `  ( x  +  1
) ) )  e. 
_V
139, 10, 11, 12ovmpt2 6209 . . . . . . . . 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 3988 . . . . . . 7  |-  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
164, 4, 15mpt2eq123i 6137 . . . . . 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 6669 . . . . . 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 5142 . . . 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 2456 . . 3  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
2120rneqi 5096 . 2  |-  ran  R  =  ran  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
221, 2, 213eqtr4i 2466 1  |-  seq  M
(  .+  ,  F
)  =  ran  R
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   omcom 4845   ran crn 4879    |` cres 4880   "cima 4881   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   reccrdg 6667   1c1 8991    + caddc 8993    seq cseq 11323
This theorem is referenced by:  seqfn  11335  seq1  11336  seqp1  11338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seq 11324
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