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Theorem frecsuc 6297
Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
Assertion
Ref Expression
frecsuc  |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S  /\  B  e. 
om )  ->  (frec ( F ,  A ) `
 suc  B )  =  ( F `  (frec ( F ,  A
) `  B )
) )
Distinct variable groups:    z, F    z, S
Allowed substitution hints:    A( z)    B( z)

Proof of Theorem frecsuc
Dummy variables  f  g  m  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 4734 . . . . . . . . 9  |-  ( f  =  g  ->  dom  f  =  dom  g )
21eqeq1d 2146 . . . . . . . 8  |-  ( f  =  g  ->  ( dom  f  =  suc  n 
<->  dom  g  =  suc  n ) )
3 fveq1 5413 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f `  n )  =  ( g `  n ) )
43fveq2d 5418 . . . . . . . . 9  |-  ( f  =  g  ->  ( F `  ( f `  n ) )  =  ( F `  (
g `  n )
) )
54eleq2d 2207 . . . . . . . 8  |-  ( f  =  g  ->  (
y  e.  ( F `
 ( f `  n ) )  <->  y  e.  ( F `  ( g `
 n ) ) ) )
62, 5anbi12d 464 . . . . . . 7  |-  ( f  =  g  ->  (
( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  <->  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `  n
) ) ) ) )
76rexbidv 2436 . . . . . 6  |-  ( f  =  g  ->  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  ( f `
 n ) ) )  <->  E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `
 n ) ) ) ) )
81eqeq1d 2146 . . . . . . 7  |-  ( f  =  g  ->  ( dom  f  =  (/)  <->  dom  g  =  (/) ) )
98anbi1d 460 . . . . . 6  |-  ( f  =  g  ->  (
( dom  f  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  y  e.  A ) ) )
107, 9orbi12d 782 . . . . 5  |-  ( f  =  g  ->  (
( E. n  e. 
om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) )  <->  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `
 n ) ) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) ) )
1110abbidv 2255 . . . 4  |-  ( f  =  g  ->  { y  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) }  =  { y  |  ( E. n  e. 
om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
1211cbvmptv 4019 . . 3  |-  ( f  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )
13 eleq1 2200 . . . . . . . 8  |-  ( y  =  x  ->  (
y  e.  ( F `
 ( g `  n ) )  <->  x  e.  ( F `  ( g `
 n ) ) ) )
1413anbi2d 459 . . . . . . 7  |-  ( y  =  x  ->  (
( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  <->  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `  n
) ) ) ) )
1514rexbidv 2436 . . . . . 6  |-  ( y  =  x  ->  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `
 n ) ) )  <->  E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) ) ) )
16 eleq1 2200 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
1716anbi2d 459 . . . . . 6  |-  ( y  =  x  ->  (
( dom  g  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  x  e.  A ) ) )
1815, 17orbi12d 782 . . . . 5  |-  ( y  =  x  ->  (
( E. n  e. 
om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) )  <->  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) ) )
1918cbvabv 2262 . . . 4  |-  { y  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) }  =  { x  |  ( E. n  e. 
om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }
2019mpteq2i 4010 . . 3  |-  ( g  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
21 suceq 4319 . . . . . . . . 9  |-  ( n  =  m  ->  suc  n  =  suc  m )
2221eqeq2d 2149 . . . . . . . 8  |-  ( n  =  m  ->  ( dom  g  =  suc  n 
<->  dom  g  =  suc  m ) )
23 fveq2 5414 . . . . . . . . . 10  |-  ( n  =  m  ->  (
g `  n )  =  ( g `  m ) )
2423fveq2d 5418 . . . . . . . . 9  |-  ( n  =  m  ->  ( F `  ( g `  n ) )  =  ( F `  (
g `  m )
) )
2524eleq2d 2207 . . . . . . . 8  |-  ( n  =  m  ->  (
x  e.  ( F `
 ( g `  n ) )  <->  x  e.  ( F `  ( g `
 m ) ) ) )
2622, 25anbi12d 464 . . . . . . 7  |-  ( n  =  m  ->  (
( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  <->  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m
) ) ) ) )
2726cbvrexv 2653 . . . . . 6  |-  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) )  <->  E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) ) )
2827orbi1i 752 . . . . 5  |-  ( ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `
 n ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) )
2928abbii 2253 . . . 4  |-  { x  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  =  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }
3029mpteq2i 4010 . . 3  |-  ( g  e.  _V  |->  { x  |  ( E. n  e.  om  ( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
3112, 20, 303eqtri 2162 . 2  |-  ( f  e.  _V  |->  { y  |  ( E. n  e.  om  ( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  \/  ( dom  f  =  (/)  /\  y  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
3231frecsuclem 6296 1  |-  ( ( A. z  e.  S  ( F `  z )  e.  S  /\  A  e.  S  /\  B  e. 
om )  ->  (frec ( F ,  A ) `
 suc  B )  =  ( F `  (frec ( F ,  A
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   _Vcvv 2681   (/)c0 3358    |-> cmpt 3984   suc csuc 4282   omcom 4499   dom cdm 4534   ` cfv 5118  freccfrec 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-recs 6195  df-frec 6281
This theorem is referenced by:  frecrdg  6298  frec2uzsucd  10167  frec2uzrdg  10175  frecuzrdgsuc  10180  frecuzrdgg  10182  frecuzrdgsuctlem  10189  seq3val  10224  seqvalcd  10225
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