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Theorem frecsuc 6126
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 4604 . . . . . . . . 9  |-  ( f  =  g  ->  dom  f  =  dom  g )
21eqeq1d 2093 . . . . . . . 8  |-  ( f  =  g  ->  ( dom  f  =  suc  n 
<->  dom  g  =  suc  n ) )
3 fveq1 5267 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f `  n )  =  ( g `  n ) )
43fveq2d 5272 . . . . . . . . 9  |-  ( f  =  g  ->  ( F `  ( f `  n ) )  =  ( F `  (
g `  n )
) )
54eleq2d 2154 . . . . . . . 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 2377 . . . . . 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 2093 . . . . . . 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 2202 . . . 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 3909 . . 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 2147 . . . . . . . 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 2377 . . . . . 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 2147 . . . . . . 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 2208 . . . 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 3900 . . 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 4203 . . . . . . . . 9  |-  ( n  =  m  ->  suc  n  =  suc  m )
2221eqeq2d 2096 . . . . . . . 8  |-  ( n  =  m  ->  ( dom  g  =  suc  n 
<->  dom  g  =  suc  m ) )
23 fveq2 5268 . . . . . . . . . 10  |-  ( n  =  m  ->  (
g `  n )  =  ( g `  m ) )
2423fveq2d 5272 . . . . . . . . 9  |-  ( n  =  m  ->  ( F `  ( g `  n ) )  =  ( F `  (
g `  m )
) )
2524eleq2d 2154 . . . . . . . 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 2587 . . . . . 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 2200 . . . 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 3900 . . 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 2109 . 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 6125 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 922    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2615   (/)c0 3275    |-> cmpt 3874   suc csuc 4166   omcom 4378   dom cdm 4411   ` cfv 4981  freccfrec 6109
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-ilim 4170  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-recs 6024  df-frec 6110
This theorem is referenced by:  frecrdg  6127  frec2uzsucd  9736  frec2uzrdg  9744  frecuzrdgsuc  9749  frecuzrdgg  9751  frecuzrdgsuctlem  9758  iseqvalt  9790
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