MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqval Unicode version

Theorem seqval 11149
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 4784 . 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 11139 . 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 2358 . . . . . . 7  |-  _V  =  _V
5 vex 2867 . . . . . . . . 9  |-  x  e. 
_V
6 vex 2867 . . . . . . . . 9  |-  y  e. 
_V
7 oveq1 5952 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
z  +  1 )  =  ( x  + 
1 ) )
87fveq2d 5612 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
98oveq2d 5961 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( x  +  1 ) ) ) )
10 oveq1 5952 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
11 eqid 2358 . . . . . . . . . 10  |-  ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
12 ovex 5970 . . . . . . . . . 10  |-  ( y 
.+  ( F `  ( x  +  1
) ) )  e. 
_V
139, 10, 11, 12ovmpt2 6070 . . . . . . . . 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 653 . . . . . . . 8  |-  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) )
1514opeq2i 3881 . . . . . . 7  |-  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
164, 4, 15mpt2eq123i 5998 . . . . . 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 6511 . . . . . 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 5033 . . . 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 2378 . . 3  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
2120rneqi 4987 . 2  |-  ran  R  =  ran  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
221, 2, 213eqtr4i 2388 1  |-  seq  M
(  .+  ,  F
)  =  ran  R
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719   omcom 4738   ran crn 4772    |` cres 4773   "cima 4774   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   reccrdg 6509   1c1 8828    + caddc 8830    seq cseq 11138
This theorem is referenced by:  seqfn  11150  seq1  11151  seqp1  11153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-seq 11139
  Copyright terms: Public domain W3C validator