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Theorem frecsuc 6078
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 4584 . . . . . . . . 9  |-  ( f  =  g  ->  dom  f  =  dom  g )
21eqeq1d 2091 . . . . . . . 8  |-  ( f  =  g  ->  ( dom  f  =  suc  n 
<->  dom  g  =  suc  n ) )
3 fveq1 5230 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f `  n )  =  ( g `  n ) )
43fveq2d 5235 . . . . . . . . 9  |-  ( f  =  g  ->  ( F `  ( f `  n ) )  =  ( F `  (
g `  n )
) )
54eleq2d 2152 . . . . . . . 8  |-  ( f  =  g  ->  (
y  e.  ( F `
 ( f `  n ) )  <->  y  e.  ( F `  ( g `
 n ) ) ) )
62, 5anbi12d 457 . . . . . . 7  |-  ( f  =  g  ->  (
( dom  f  =  suc  n  /\  y  e.  ( F `  (
f `  n )
) )  <->  ( dom  g  =  suc  n  /\  y  e.  ( F `  ( g `  n
) ) ) ) )
76rexbidv 2375 . . . . . 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 2091 . . . . . . 7  |-  ( f  =  g  ->  ( dom  f  =  (/)  <->  dom  g  =  (/) ) )
98anbi1d 453 . . . . . 6  |-  ( f  =  g  ->  (
( dom  f  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  y  e.  A ) ) )
107, 9orbi12d 740 . . . . 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 2200 . . . 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 3894 . . 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 2145 . . . . . . . 8  |-  ( y  =  x  ->  (
y  e.  ( F `
 ( g `  n ) )  <->  x  e.  ( F `  ( g `
 n ) ) ) )
1413anbi2d 452 . . . . . . 7  |-  ( y  =  x  ->  (
( dom  g  =  suc  n  /\  y  e.  ( F `  (
g `  n )
) )  <->  ( dom  g  =  suc  n  /\  x  e.  ( F `  ( g `  n
) ) ) ) )
1514rexbidv 2375 . . . . . 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 2145 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
1716anbi2d 452 . . . . . 6  |-  ( y  =  x  ->  (
( dom  g  =  (/) 
/\  y  e.  A
)  <->  ( dom  g  =  (/)  /\  x  e.  A ) ) )
1815, 17orbi12d 740 . . . . 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 2206 . . . 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 3886 . . 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 4186 . . . . . . . . 9  |-  ( n  =  m  ->  suc  n  =  suc  m )
2221eqeq2d 2094 . . . . . . . 8  |-  ( n  =  m  ->  ( dom  g  =  suc  n 
<->  dom  g  =  suc  m ) )
23 fveq2 5231 . . . . . . . . . 10  |-  ( n  =  m  ->  (
g `  n )  =  ( g `  m ) )
2423fveq2d 5235 . . . . . . . . 9  |-  ( n  =  m  ->  ( F `  ( g `  n ) )  =  ( F `  (
g `  m )
) )
2524eleq2d 2152 . . . . . . . 8  |-  ( n  =  m  ->  (
x  e.  ( F `
 ( g `  n ) )  <->  x  e.  ( F `  ( g `
 m ) ) ) )
2622, 25anbi12d 457 . . . . . . 7  |-  ( n  =  m  ->  (
( dom  g  =  suc  n  /\  x  e.  ( F `  (
g `  n )
) )  <->  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m
) ) ) ) )
2726cbvrexv 2584 . . . . . 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 713 . . . . 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 2198 . . . 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 3886 . . 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 2107 . 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 6077 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 102    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2611   (/)c0 3268    |-> cmpt 3860   suc csuc 4149   omcom 4360   dom cdm 4392   ` cfv 4953  freccfrec 6061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-recs 5976  df-frec 6062
This theorem is referenced by:  frecrdg  6079  frec2uzsucd  9560  frec2uzrdg  9568  frecuzrdgsuc  9573  frecuzrdgg  9575  frecuzrdgsuctlem  9582  iseqvalt  9609
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