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Theorem seqval 12086
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 5012 . 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 12076 . 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 2467 . . . . . . 7  |-  _V  =  _V
5 vex 3116 . . . . . . . . 9  |-  x  e. 
_V
6 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
7 oveq1 6291 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
z  +  1 )  =  ( x  + 
1 ) )
87fveq2d 5870 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
98oveq2d 6300 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( x  +  1 ) ) ) )
10 oveq1 6291 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
11 eqid 2467 . . . . . . . . . 10  |-  ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
12 ovex 6309 . . . . . . . . . 10  |-  ( y 
.+  ( F `  ( x  +  1
) ) )  e. 
_V
139, 10, 11, 12ovmpt2 6422 . . . . . . . . 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 672 . . . . . . . 8  |-  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) )
1514opeq2i 4217 . . . . . . 7  |-  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
164, 4, 15mpt2eq123i 6344 . . . . . 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 7077 . . . . . 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 5 . . . . 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 5269 . . . 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 2496 . . 3  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
2120rneqi 5229 . 2  |-  ran  R  =  ran  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
221, 2, 213eqtr4i 2506 1  |-  seq M
(  .+  ,  F
)  =  ran  R
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   ran crn 5000    |` cres 5001   "cima 5002   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   omcom 6684   reccrdg 7075   1c1 9493    + caddc 9495    seqcseq 12075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-recs 7042  df-rdg 7076  df-seq 12076
This theorem is referenced by:  seqfn  12087  seq1  12088  seqp1  12090
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